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== 4.2#h(0.6em)Graphs of Trigonometric Functions


    
      
=== A Periodic Function of Angle


      Imagine that you are riding on a Ferris wheel. As the wheel turns, your height above the ground increases and then decreases again, repeating the same pattern each time the Ferris wheel makes a complete rotation. This pattern is an example of a #strong[periodic function]. We use periodic functions to model phenomena that exhibit cyclical behavior, such as the height of tides, seasonal patterns of growth in plants and animals, radio waves, and planetary motion.


      The Ferris wheel this paragraph imagines: a wheel of radius 10 meters whose hub sits 11 meters up. Drag the angle slider to ride: your height above the ground is 11 + 10 sin(a°), climbing to 21 meters at the top of the wheel (a = 90°) and dipping to 1 meter at the bottom (a = 270°). Keep dragging past 360° — the same heights return in the same order, once each full turn. That repetition, the same output every 360° of input, is what the section goes on to name a periodic function, and it is the reason the graph of sin θ you are about to meet is a wave.{"functions":\[{"color":"\#6a6a6a","expression":"10\*cos(t)","expression2":"11 + 10\*sin(t)","id":"wheel","inequality":"lt","lineDash":\[\],"lineWidth":2,"mode":"parametric","tMax":6.2832,"tMin":0,"variable":"x","visible":true},{"color":"\#1f4e79","expression":"t\*10\*cos(a\*pi/180)","expression2":"11 + t\*10\*sin(a\*pi/180)","id":"spoke","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"parametric","tMax":1,"tMin":0,"variable":"x","visible":true},{"color":"\#b8442a","expression":"10\*cos(a\*pi/180) + 0.7\*cos(t)","expression2":"11 + 10\*sin(a\*pi/180) + 0.7\*sin(t)","id":"rider","inequality":"lt","lineDash":\[\],"lineWidth":2.5,"mode":"parametric","tMax":6.2832,"tMin":0,"variable":"x","visible":true},{"color":"\#b8442a","expression":"10\*cos(a\*pi/180)","expression2":"t\*(11 + 10\*sin(a\*pi/180))","id":"height","inequality":"lt","lineDash":\[6,4\],"lineWidth":2,"mode":"parametric","tMax":1,"tMin":0,"variable":"x","visible":true}\],"grid":{"majorSpacing":5,"showAxes":true,"showLabels":true},"parameters":\[{"id":"p-a","label":"Angle turned","max":720,"min":0,"name":"a","step":5,"unit":"\\u00b0","value":30}\],"title":"Ferris wheel: your height as the wheel turns","version":1,"viewport":{"centerX":0,"centerY":11,"scale":19}}We'll create a #strong[mathematical model] for a ride on a Ferris wheel that has a radius of 100 feet and rotates counterclockwise. Our model will be a function that describes your height above the ground as you ride the wheel. In order to graph the Ferris wheel function, we must first specify the input and output variables, and then choose a coordinate system to display their values.


      We'll place the origin at the center of the Ferris wheel. Then the line from the origin to your position on the wheel makes an angle with the horizontal, as shown at right.  This angle, #math.equation(block: false, alt: "θ")[$θ$], will be the #strong[input variable] for the function. Your height, #math.equation(block: false, alt: "h")[$h$], is also a variable, and is related to the #math.equation(block: false, alt: "y")[$y$]-coordinate of your position; in fact, we see that #math.equation(block: false, alt: "h equals y plus 100")[$h = y + 100$], because the center of the wheel is 100 feet above the ground.


      #figure(figph[ferris wheel], alt: "ferris wheel", caption: none)


      To simplify the model, we'll first graph #math.equation(block: false, alt: "y")[$y$] as the #strong[output variable], instead of #math.equation(block: false, alt: "h")[$h$]. As the angle #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], your #math.equation(block: false, alt: "y")[$y$]-coordinate increases from 0 to 100. You are then at the top of the wheel. Then, as #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] to #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], your #math.equation(block: false, alt: "y")[$y$]-coordinate decreases from 100 back to 0.


      #figure(figph[ferris wheel and sine function], alt: "ferris wheel and sine function", caption: none)


      Finally, as #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], your #math.equation(block: false, alt: "y")[$y$]-coordinate decreases from 0 to #math.equation(block: false, alt: "minus 100")[$− 100$] and then increases from #math.equation(block: false, alt: "minus 100")[$− 100$] back to 0. You have made one complete rotation on the Ferris wheel. If you go around again, #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] to #math.equation(block: false, alt: "720 degrees")[$720^(∘)$], and the graph of your #math.equation(block: false, alt: "y")[$y$]-coordinate will repeat the pattern of the first rotation. The figure above shows how your #math.equation(block: false, alt: "y")[$y$]-coordinate is plotted as a function of the angle #math.equation(block: false, alt: "θ")[$θ$].


      Look back at the diagram of the Ferris wheel and notice that #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis equals the fraction y over 100")[$" " sin ( θ ) = display(frac(y, 100))$], so


      #math.equation(block: true, alt: "y equals 100 sin ⁡ open parenthesis θ close parenthesis")[$y = 100 sin ( θ )$]
      For example, when #math.equation(block: false, alt: "θ equals 30 degrees")[$θ = 30^(∘)$], the #math.equation(block: false, alt: "y")[$y$]-coordinate is


      #math.equation(block: true, alt: "y equals 100 sin ⁡ open parenthesis 30 degrees close parenthesis equals 100 open parenthesis the fraction 1 over 2 close parenthesis equals 50")[$y = 100 sin ( 30^(∘) ) = 100 ( display(frac(1, 2)) ) = 50$]
      and your height above the ground is


      #math.equation(block: true, alt: "h equals y plus 100 equals 150 feet")[$h = y + 100 = 150 " " " " "feet"$]
      #figure(figph[ferris wheel at 30 degrees], alt: "ferris wheel at 30 degrees", caption: none)


      In general, then, #math.equation(block: false, alt: "h")[$h$] is given as a function of #math.equation(block: false, alt: "θ")[$θ$] by


      #math.equation(block: true, alt: "h equals y plus 100 equals 100 sin ⁡ open parenthesis θ close parenthesis plus 100")[$h = y + 100 = 100 sin ( θ ) + 100$]
      This is our model for your height on a Ferris wheel ride.


    
    
      
=== The Sine Function


      Our Ferris wheel model used values of #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$], so let us explore its properties. Remember that the trigonometric ratio #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$] is actually a function of the angle #math.equation(block: false, alt: "θ")[$θ$]. Thus, for each value of #math.equation(block: false, alt: "θ")[$θ$], there is only one value of #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$], and we may write #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$].


      #notebox("Activity", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Graph of the Sine Function]


        Graph the sine function #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$" " f ( θ ) = sin ( θ )$].


        + First make a table of values. You can use your calculator's table feature to fill in the following values, rounded to two decimal places.#figure(table(
  columns: 11, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "10 degrees")[$10^(∘)$]], [#math.equation(block: false, alt: "20 degrees")[$20^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "40 degrees")[$40^(∘)$]], [#math.equation(block: false, alt: "50 degrees")[$50^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "70 degrees")[$70^(∘)$]], [#math.equation(block: false, alt: "80 degrees")[$80^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$],
))
+ Plot the points in the table and connect them with a smooth curve to graph the sine function in the first quadrant, from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$].
+ Recall that the values of #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$] in the second quadrant can be found using reference angles: #math.equation(block: false, alt: "sin ⁡ open parenthesis 180 degrees minus θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$sin ( 180^(∘) − θ ) = sin ( θ )$].  This fact gives us values of #math.equation(block: false, alt: "sin ⁡ θ")[$sin θ$]  from #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] to #math.equation(block: false, alt: "180 degrees")[$180^(∘)$].#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "100 degrees")[$100^(∘)$]], [#math.equation(block: false, alt: "110 degrees")[$110^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "130 degrees")[$130^(∘)$]], [#math.equation(block: false, alt: "140 degrees")[$140^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "160 degrees")[$160^(∘)$]], [#math.equation(block: false, alt: "170 degrees")[$170^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$],
))

 #linebreak() Continue your graph from part (2) for second quadrant angles.
+ The values of #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$] in the third and fourth quadrants are the negatives of their values in the second and first quadrants.#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "190 degrees")[$190^(∘)$]], [#math.equation(block: false, alt: "200 degrees")[$200^(∘)$]], [#math.equation(block: false, alt: "210 degrees")[$210^(∘)$]], [#math.equation(block: false, alt: "220 degrees")[$220^(∘)$]], [#math.equation(block: false, alt: "230 degrees")[$230^(∘)$]], [#math.equation(block: false, alt: "240 degrees")[$240^(∘)$]], [#math.equation(block: false, alt: "250 degrees")[$250^(∘)$]], [#math.equation(block: false, alt: "260 degrees")[$260^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$],
))

#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "280 degrees")[$280^(∘)$]], [#math.equation(block: false, alt: "290 degrees")[$290^(∘)$]], [#math.equation(block: false, alt: "300 degrees")[$300^(∘)$]], [#math.equation(block: false, alt: "310 degrees")[$310^(∘)$]], [#math.equation(block: false, alt: "320 degrees")[$320^(∘)$]], [#math.equation(block: false, alt: "330 degrees")[$330^(∘)$]], [#math.equation(block: false, alt: "340 degrees")[$340^(∘)$]], [#math.equation(block: false, alt: "350 degrees")[$350^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$],
))
+ The graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$" " f ( θ ) = sin ( θ ) " "$] from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] is shown below.#figure(figph[sine graph], alt: "sine graph", caption: none)
]


      If we continue the graph for angles larger than #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] or smaller than #math.equation(block: false, alt: "0 degrees")[$0^(∘)$], we find that the same pattern repeats, as shown below. This should not be surprising, because we know that coterminal angles have the same trigonometric ratios.


      #figure(figph[sine graph], alt: "sine graph", caption: none)


      The sine is an example of a periodic function. The smallest interval on which the graph repeats is called the #strong[period] of the graph. From the graph in the previous example, we make the following observations:


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Properties of the Sine Function]


        - The #strong[period] of the sine function is #math.equation(block: false, alt: "360 degrees")[$360^(∘)$].
- The maximum and minimum function values are 1 and 1, respectively.
- The graph oscillates around its #strong[midline], the horizontal line #math.equation(block: false, alt: "y equals 0")[$y = 0$].
- The distance between the midline and either the maximum or minimum value is called the #strong[amplitude] of the function, so the amplitude of the sine function is 1.
]


      You can use your calculator to graph the sine function, by entering


      #math.equation(block: false, alt: "Y sub 1 equals")[$#h(2em) #h(2em) #h(2em) "Y"_(1) =$]SIN#math.equation(block: false, alt: "X , T , θ , n")[$" " #box(stroke: 0.6pt, inset: 3pt, outset: 0pt)[$display(X "," T "," θ "," n)$]$]


      and pressing  ZOOM #math.equation(block: false, alt: "7")[$7$] for the trig window. The graph shows two periods of the sine function, from  #math.equation(block: false, alt: "θ equals minus 360 degrees")[$θ = − 360^(∘)$] to #math.equation(block: false, alt: "θ equals 360 degrees")[$θ = 360^(∘)$].


      
        Your height #math.equation(block: false, alt: "h")[$h$] on the Ferris wheel is a function of #math.equation(block: false, alt: "θ")[$θ$],


        #math.equation(block: true, alt: "h equals 100 plus y equals 100 plus 100 sin ⁡ open parenthesis θ close parenthesis")[$h = 100 + y = 100 + 100 sin ( θ )$]
        + Complete the table of values and graph the Ferris wheel function, #math.equation(block: false, alt: "h equals F open parenthesis θ close parenthesis")[$h = F ( θ )$].
        #figure(table(
  columns: 8, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [#math.equation(block: false, alt: "h equals F open parenthesis θ close parenthesis")[$h = F ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))

#figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "210 degrees")[$210^(∘)$]], [#math.equation(block: false, alt: "240 degrees")[$240^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "300 degrees")[$300^(∘)$]], [#math.equation(block: false, alt: "330 degrees")[$330^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [#math.equation(block: false, alt: "h equals F open parenthesis θ close parenthesis")[$h = F ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))

#figure(figph[grid], alt: "grid", caption: none)
+ Give the period, amplitude, and midline of the graph.


        
          + #figure(table(
  columns: 8, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "0.866")[$0.866$]], [#math.equation(block: false, alt: "1.0")[$1.0$]], [#math.equation(block: false, alt: "0.866")[$0.866$]], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "0")[$0$]],
  [#math.equation(block: false, alt: "h equals F open parenthesis θ close parenthesis")[$h = F ( θ )$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "186.6")[$186.6$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "186.6")[$186.6$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "100")[$100$]],
))

#figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "210 degrees")[$210^(∘)$]], [#math.equation(block: false, alt: "240 degrees")[$240^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "300 degrees")[$300^(∘)$]], [#math.equation(block: false, alt: "330 degrees")[$330^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [#math.equation(block: false, alt: "minus 0.5")[$− 0.5$]], [#math.equation(block: false, alt: "minus 0.866")[$− 0.866$]], [#math.equation(block: false, alt: "minus 1.0")[$− 1.0$]], [#math.equation(block: false, alt: "minus 0.866")[$− 0.866$]], [#math.equation(block: false, alt: "minus 0.5")[$− 0.5$]], [#math.equation(block: false, alt: "0")[$0$]],
  [#math.equation(block: false, alt: "h equals F open parenthesis θ close parenthesis")[$h = F ( θ )$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "13.4")[$13.4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "13.4")[$13.4$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "100")[$100$]],
))

#figure(figph[sine graph], alt: "sine graph", caption: none)
+ Period: #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], amplitude: 100, midline: #math.equation(block: false, alt: "h equals 100")[$h = 100$]


        
      
    
    
      
=== The Cosine Function


      In the previous exercise you graphed the height of a person riding on a Ferris wheel. Your graph involved #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$], because the sine function tells us the #math.equation(block: false, alt: "y")[$y$]-coordinate of a point that travels around a circle. The cosine function tells us the #math.equation(block: false, alt: "x")[$x$]-coordinate of a point that travels around a circle.


      #examplebox("Example 1")[][
Small adjustments to the fit of a bicycle can affect both the cyclist's efficiency and the stress on his or her joints.  The KOPS rule (Knee Over Pedal Stem) aligns the cyclist's knee directly over the pedal at the point of maximum force, as shown at right.  As the cyclist's foot rotates away from this KOPS line, stress on the knee increases.


          #figure(figph[bicycle], alt: "bicycle", caption: none)


          + Suppose the pedal crank is 18 centimeters long.  When the crank makes an angle #math.equation(block: false, alt: "θ")[$θ$] with the horizontal, how far is the cyclist's foot displaced horizontally from the KOPS line?
+ Graph the horizontal displacement, #math.equation(block: false, alt: "d")[$d$], as a function of #math.equation(block: false, alt: "θ")[$θ$].


        
        #solutionbox[
+ The cyclist's foot travels around a circle of radius 18 centimeters. If we place the origin at the center of the chain gear, the #math.equation(block: false, alt: "x")[$x$]-coordinate of the foot is given by 
            #math.equation(block: true, alt: "x equals r cos ⁡ open parenthesis θ close parenthesis equals 18 cos ⁡ open parenthesis θ close parenthesis")[$x = r cos ( θ ) = 18 cos ( θ )$] (See the figure at right.) The KOPS line is the vertical line #math.equation(block: false, alt: "x equals 18")[$x = 18$], so the horizontal distance between the cyclist's foot and the KOPS line is

#math.equation(block: true, alt: "d equals 18 minus 18 cos ⁡ open parenthesis θ close parenthesis")[$d = 18 − 18 cos ( θ )$]#figure(figph[KOPS line and circle], alt: "KOPS line and circle", caption: none)
+ You can use your calculator to verify the table and graph for the function 
        #math.equation(block: true, alt: "d equals 18 minus 18 cos ⁡ open parenthesis θ close parenthesis")[$d = 18 − 18 cos ( θ )$] 
        shown below.#figure(table(
  columns: 14, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "210 degrees")[$210^(∘)$]], [#math.equation(block: false, alt: "240 degrees")[$240^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "300 degrees")[$300^(∘)$]], [#math.equation(block: false, alt: "330 degrees")[$330^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "d")[$d$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2.4")[$2.4$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "33.6")[$33.6$]], [#math.equation(block: false, alt: "36")[$36$]], [#math.equation(block: false, alt: "33.6")[$33.6$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "2.4")[$2.4$]], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(figph[cosine graph], alt: "cosine graph", caption: none)
]
]


      In the example above, the period is #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], the amplitude is 18 cm, and the midline is #math.equation(block: false, alt: "d equals 18")[$d = 18$].


      You can see that the cosine graph is similar to the sine graph, but they are not identical.


      
        + Complete the table below with values rounded to two decimal places. Use the table and your knowledge of reference angles to graph the cosine function, #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis ,")[$" " f ( θ ) = cos ( θ ) , " "$] from #math.equation(block: false, alt: "minus 180 degrees")[$− 180^(∘)$] to #math.equation(block: false, alt: "540 degrees")[$540^(∘)$].#figure(table(
  columns: 11, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "10 degrees")[$10^(∘)$]], [#math.equation(block: false, alt: "20 degrees")[$20^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "40 degrees")[$40^(∘)$]], [#math.equation(block: false, alt: "50 degrees")[$50^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "70 degrees")[$70^(∘)$]], [#math.equation(block: false, alt: "80 degrees")[$80^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]]),
  [#math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))

#figure(figph[grid], alt: "grid", caption: none)
+ Use your graph to find the period, amplitude, and midline of the cosine function. How does the graph of cosine differ from the graph of sine? (Hint:  Consider the intercepts of the graph, and the location of the maximum and minimum values.)


        
          + #figure(table(
  columns: 11, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "10 degrees")[$10^(∘)$]], [#math.equation(block: false, alt: "20 degrees")[$20^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "40 degrees")[$40^(∘)$]], [#math.equation(block: false, alt: "50 degrees")[$50^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "70 degrees")[$70^(∘)$]], [#math.equation(block: false, alt: "80 degrees")[$80^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]]),
  [#math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0.98")[$0.98$]], [#math.equation(block: false, alt: "0.94")[$0.94$]], [#math.equation(block: false, alt: "0.87")[$0.87$]], [#math.equation(block: false, alt: "0.77")[$0.77$]], [#math.equation(block: false, alt: "0.64")[$0.64$]], [#math.equation(block: false, alt: "0.50")[$0.50$]], [#math.equation(block: false, alt: "0.34")[$0.34$]], [#math.equation(block: false, alt: "0.17")[$0.17$]], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(figph[cosine graph], alt: "cosine graph", caption: none)
+ Period: #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], amplitude: #math.equation(block: false, alt: "1")[$1$],  midline: #math.equation(block: false, alt: "y equals 0")[$y = 0$].  The cosine graph starts #math.equation(block: false, alt: "open parenthesis θ equals 0 degrees close parenthesis")[$( θ = 0^(∘) )$] at its high point, while the sine graph starts #math.equation(block: false, alt: "open parenthesis θ equals 0 degrees close parenthesis")[$( θ = 0^(∘) )$] at its midline.


        
      
    
    
      
=== Interlude: Review of Function Notation


      Perhaps it is time to review our use of function notation. Recall that we use the notation #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$" " y = f ( x ) " "$] to indicate that #math.equation(block: false, alt: "y")[$y$] is a function of #math.equation(block: false, alt: "x")[$x$], that is, #math.equation(block: false, alt: "x")[$x$] is the input variable and #math.equation(block: false, alt: "y")[$y$] is the output variable.


      #examplebox("Example 2")[][
Make a table of input and output values and a graph for the function


          #math.equation(block: true, alt: "y equals f open parenthesis x close parenthesis equals the square root of 9 minus x squared")[$y = f ( x ) = sqrt(9 − x^(2))$]
        
        #solutionbox[
We choose several values for the input variable, #math.equation(block: false, alt: "x")[$x$], and evaluate the function to find the corresponding values of the output variable, #math.equation(block: false, alt: "y")[$y$]. For example,


          #math.equation(block: true, alt: "f open parenthesis minus 3 close parenthesis equals the square root of 9 minus open parenthesis minus 3 close parenthesis squared equals 0")[$f ( − 3 ) = sqrt(9 − ( − 3 )^(2)) = 0$]
          We plot the points in the table and connect them to obtain the graph shown at right.


          #figure(table(
  columns: 8, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$]], [#math.equation(block: false, alt: "2 the square root of 2")[$2 sqrt(2)$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2 the square root of 2")[$2 sqrt(2)$]], [#math.equation(block: false, alt: "the square root of 5")[$sqrt(5)$]], [#math.equation(block: false, alt: "0")[$0$]],
))


          #figure(figph[semicircle graph], alt: "semicircle graph", caption: none)
]
]


      Of course, we don't always use #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] for the input and output variables.  In the previous example, we could write #math.equation(block: false, alt: "w equals f open parenthesis t close parenthesis equals the square root of 9 minus t squared")[$" " w = f ( t ) = sqrt(9 − t^(2)) " "$] for the function, so that #math.equation(block: false, alt: "t")[$t$] is the input and #math.equation(block: false, alt: "w")[$w$] is the output. The table of values and the graph are the same; only the names of the variables have changed.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
When we discuss trigonometric functions, there are several variables involved. Our definitions of the trig ratios involve four variables: #math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y")[$y$], #math.equation(block: false, alt: "r")[$r$], and #math.equation(block: false, alt: "θ")[$θ$], as illustrated below.


        #figure(figph[angles], alt: "angles", caption: none)


        #figure(figph[graphs], alt: "graphs", caption: none)


        If the value of #math.equation(block: false, alt: "r")[$r$] is fixed for a given situation, such as the Ferris wheel or the bicycle wheel discussed above, then #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] are both functions of #math.equation(block: false, alt: "θ")[$θ$]. This means that the values of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] depend only on the value of the angle #math.equation(block: false, alt: "θ")[$θ$].  If #math.equation(block: false, alt: "r equals 1")[$r = 1$], we have


        #math.equation(block: true, alt: "x, equals f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis; y, equals g open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$x & = f ( θ ) = cos ( θ ) \ y & = g ( θ ) = sin ( θ )$]
        The graphs of these functions are shown below. Note particularly that the horizontal axis displays values of the input variable, and the vertical axis displays the output variable.


        #figure(figph[cosine graph], alt: "cosine graph", caption: none)


        #figure(figph[sine graph], alt: "sine graph", caption: none)


        If we use different variables for the input and output, the functions and their graphs are the same, but the axes should be labeled with the appropriate variables.
]


      
        Sketch a graph of each function, and label the axes.


        + #math.equation(block: false, alt: "d equals F open parenthesis ϕ close parenthesis equals sin ⁡ open parenthesis ϕ close parenthesis")[$d = F ( ϕ ) = sin ( ϕ )$]
+ #math.equation(block: false, alt: "t equals G open parenthesis β close parenthesis equals cos ⁡ open parenthesis β close parenthesis")[$t = G ( β ) = cos ( β )$]


        
          In particular, we label the axes with the given variables.


          + #figure(figph[sine graph], alt: "sine graph", caption: none)
+ #figure(figph[cosine graph], alt: "cosine graph", caption: none)


        
      
      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
When we write #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$], we are using "cos" as the name of a function whose input is #math.equation(block: false, alt: "θ")[$θ$].


        A common mistake is to think of #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$] as a product, cos #emph[times] #math.equation(block: false, alt: "θ")[$θ$], but this makes no sense, because "cos" by itself has no meaning. Remember that #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$]  represents a single number, namely the output of the cosine function.
]


    
    
      
=== The Tangent Function


      The tangent function is periodic, but its graph is not similar to the graphs of sine and cosine. Recall that the tangent of an angle in standard position is defined by


      #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis equals the fraction y over x")[$tan ( θ ) = display(frac(y, x))$]
      Study the figure at right to see that as #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], #math.equation(block: false, alt: "y")[$y$] increases while #math.equation(block: false, alt: "x")[$x$] remains constant, so the value of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] increases.


      #figure(figph[Right triangles with a common base x and increasing y], alt: "Right triangles with a common base x and increasing y", caption: none)


      #examplebox("Example 3")[][
Sketch a graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals tan ⁡ open parenthesis θ close parenthesis")[$" " f ( θ ) = tan ( θ ) " "$] for #math.equation(block: false, alt: "0 degrees less than or equal to θ less than or equal to 180 degrees")[$0^(∘) ≤ θ ≤ 180^(∘)$].


        
        #solutionbox[
You can use your calculator to verify the following values for #math.equation(block: false, alt: "tan ⁡ θ")[$tan θ$].


          #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "70 degrees")[$70^(∘)$]], [#math.equation(block: false, alt: "80 degrees")[$80^(∘)$]], [#math.equation(block: false, alt: "85 degrees")[$85^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.58")[$0.58$]], [#math.equation(block: false, alt: "1.73")[$1.73$]], [#math.equation(block: false, alt: "2.75")[$2.75$]], [#math.equation(block: false, alt: "5.67")[$5.67$]], [#math.equation(block: false, alt: "11.43")[$11.43$]],
))


          As #math.equation(block: false, alt: "θ")[$θ$] gets closer to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] increases very rapidly. Recall that #math.equation(block: false, alt: "tan ⁡ open parenthesis 90 degrees close parenthesis")[$tan ( 90^(∘) )$] is undefined, so there is no point on the graph at #math.equation(block: false, alt: "θ equals 90 degrees")[$θ = 90^(∘)$]. The graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals tan ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = tan ( θ )$] for #math.equation(block: false, alt: "0 less than or equal to θ less than 90 degrees")[$0 ≤ θ < 90^(∘)$] is shown at right.


          #figure(figph[graph of tangent in first quadrant], alt: "graph of tangent in first quadrant", caption: none)


          For #math.equation(block: false, alt: "θ")[$θ$] in the second quadrant, the tangent is negative. The reference angle for each angle in the second quadrant is its supplement, so


          #math.equation(block: true, alt: "tan ⁡ open parenthesis θ close parenthesis equals minus tan ⁡ open parenthesis 180 degrees minus θ close parenthesis")[$tan ( θ ) = − tan ( 180^(∘) − θ )$]
          as shown at right. For example, you can verify that


          #figure(figph[supplementary angles], alt: "supplementary angles", caption: none)


          #math.equation(block: true, alt: "tan ⁡ open parenthesis 130 degrees close parenthesis equals minus tan ⁡ open parenthesis 180 degrees minus 130 degrees close parenthesis equals minus tan ⁡ open parenthesis 50 degrees close parenthesis equals minus 1.19")[$tan ( 130^(∘) ) = − tan ( 180^(∘) − 130^(∘) ) = − tan ( 50^(∘) ) = − 1.19$]
          In particular, for values of #math.equation(block: false, alt: "θ")[$θ$] close to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], the values of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] are large negative numbers. We plot several points and sketch the graph in the second quadrant.


          #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "100 degrees")[$100^(∘)$]], [#math.equation(block: false, alt: "110 degrees")[$110^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [#math.equation(block: false, alt: "minus 5.67")[$− 5.67$]], [#math.equation(block: false, alt: "minus 2.75")[$− 2.75$]], [#math.equation(block: false, alt: "minus 1.73")[$− 1.73$]], [#math.equation(block: false, alt: "minus 0.58")[$− 0.58$]], [#math.equation(block: false, alt: "0")[$0$]],
))


          In the figure at right, note that the graph has a break at #math.equation(block: false, alt: "θ equals 90 degrees")[$θ = 90^(∘)$], because  #math.equation(block: false, alt: "tan ⁡ open parenthesis 90 degrees close parenthesis")[$tan ( 90^(∘) )$] is undefined.


          #figure(figph[tangent graph], alt: "tangent graph", caption: none)
]
]


      Now let's consider the graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals tan ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = tan ( θ )$] in the third and fourth quadrants. The tangent is positive in the third quadrant, and negative in the fourth quadrant. In fact, from the figure below you can see that the angles #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "180 degrees plus θ")[$180^(∘) + θ$] are vertical angles.


      #figure(figph[angles differing by 180 degrees], alt: "angles differing by 180 degrees", caption: none)


      #figure(figph[angles differing by 180 deg], alt: "angles differing by 180 deg", caption: none)


      Because #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "180 degrees plus θ")[$180^(∘) + θ$] have the same reference angle, they have the same tangent.  For example,


      #math.equation(block: true, alt: "tan ⁡ open parenthesis 200 degrees close parenthesis, equals tan ⁡ open parenthesis 20 degrees close parenthesis; tan ⁡ open parenthesis 230 degrees close parenthesis, equals tan ⁡ open parenthesis 50 degrees close parenthesis; tan ⁡ open parenthesis 250 degrees close parenthesis, equals tan ⁡ open parenthesis 70 degrees close parenthesis")[$tan ( 200^(∘) ) & = tan ( 20^(∘) ) \ tan ( 230^(∘) ) & = tan ( 50^(∘) ) \ tan ( 250^(∘) ) & = tan ( 70^(∘) )$]
      Thus, the graph of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] in the third quadrant is the same as its graph in the first quadrant. Similarly, the graph of the tangent function in the fourth quadrant is the same as its graph in the second quadrant. The completed graph is shown below.


      #figure(figph[tangent graph], alt: "tangent graph", caption: none)


      
        + What is the period of the tangent function?
+ Does the graph of tangent have an amplitude?
+ For what values of #math.equation(block: false, alt: "θ")[$θ$] is #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] undefined?
+ Give the equations of any horizontal or vertical asymptotes for #math.equation(block: false, alt: "0 degrees less than or equal to θ less than or equal to 360 degrees")[$0^(∘) ≤ θ ≤ 360^(∘)$].


        
          + #math.equation(block: false, alt: "180 degrees")[$180^(∘)$]
+ No
+ #math.equation(block: false, alt: "90 degrees , 270 degrees")[$90^(∘) , " " 270^(∘)$], and their coterminal angles
+ The graph has vertical asymptotes at #math.equation(block: false, alt: "θ equals 90 degrees")[$θ = 90^(∘)$] and #math.equation(block: false, alt: "θ equals 270 degrees")[$θ = 270^(∘)$]


        
      
    
    
      
=== Period, Midline and Amplitude


      All sine and cosine graphs have the characteristic "wave" shape we've seen in previous examples. But we can alter the size and frequency of the waves by changing the formula for the function. In the next example we consider three variations of the sine function.


      #examplebox("Example 4")[][
Make a table of values and sketch a graph for each of the functions. How does each differ from the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis θ close parenthesis")[$" " y = sin ( θ ) " "$] ?


          + #math.equation(block: false, alt: "y equals 3 sin ⁡ open parenthesis θ close parenthesis")[$y = 3 sin ( θ )$]
+ #math.equation(block: false, alt: "y equals 3 plus sin ⁡ open parenthesis θ close parenthesis")[$y = 3 + sin ( θ )$]
+ #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis 3 θ close parenthesis")[$y = sin ( 3 θ )$]


        
        #solutionbox[
+ We make a table with multiples of #math.equation(block: false, alt: "45 degrees")[$45^(∘)$]. #linebreak() #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "y equals 3 sin ⁡ open parenthesis θ close parenthesis")[$y = 3 sin ( θ )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2.1")[$2.1$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2.1")[$2.1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 2.1")[$− 2.1$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2.1")[$− 2.1$]], [#math.equation(block: false, alt: "0")[$0$]],
))

We plot the points, and connect them with a sine-shaped wave. Compare the graph, shown at right, to the graph of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis θ close parenthesis")[$y = sin ( θ )$]. The graph is like a sine graph, except that it oscillates between a maximum value of #math.equation(block: false, alt: "3")[$3$] and a minimum value of #math.equation(block: false, alt: "minus 3")[$− 3$]. The amplitude of this function is #math.equation(block: false, alt: "3")[$3$].

#figure(figph[graphs of sine and 3 sine], alt: "graphs of sine and 3 sine", caption: none)
+ Again, we make a table of values with multiples of #math.equation(block: false, alt: "45 degrees")[$45^(∘)$] and plot the points. #linebreak() #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "y equals 3 plus sin ⁡ open parenthesis θ close parenthesis")[$y = 3 + sin ( θ )$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "3.7")[$3.7$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3.7")[$3.7$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2.3")[$2.3$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "2.1")[$2.1$]], [#math.equation(block: false, alt: "3")[$3$]],
))

This graph has the same amplitude as #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis θ close parenthesis")[$y = sin ( θ )$], but the entire graph is shifted up by #math.equation(block: false, alt: "3")[$3$] units, as shown at right. The midline of this function is the line #math.equation(block: false, alt: "y equals 3")[$y = 3$].

#figure(figph[graph of sine and translated sine], alt: "graph of sine and translated sine", caption: none)
+ This time we'll make a table with multiples of #math.equation(block: false, alt: "15 degrees")[$15^(∘)$]. #linebreak() #figure(table(
  columns: 13, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "15 degrees")[$15^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "75 degrees")[$75^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "105 degrees")[$105^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "165 degrees")[$165^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]]),
  [#math.equation(block: false, alt: "y equals sin ⁡ open parenthesis 3 θ close parenthesis")[$y = sin ( 3 θ )$]], [#math.equation(block: false, alt: "0.7")[$0.7$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0.7")[$0.7$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 0.7")[$− 0.7$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 0.7")[$− 0.7$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.7")[$0.7$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0.7")[$0.7$]], [#math.equation(block: false, alt: "0")[$0$]],
))

You can continue the table for #math.equation(block: false, alt: "θ")[$θ$] between #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], and plot the points to find the graph shown at right. The graph has the same amplitude and midline as #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis θ close parenthesis")[$y = sin ( θ )$], but it completes three cycles from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] instead of one cycle. The period of this graph is one-third of #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], or #math.equation(block: false, alt: "120 degrees")[$120^(∘)$].

#figure(figph[graph of sine and sine 3 theta], alt: "graph of sine and sine 3 theta", caption: none)
]
]


      The graphs in the previous example illustrate a general rule about sine and cosine graphs.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Amplitude, Period, and Midline]


        + The graph of #math.equation(block: true, alt: "y equals A cos ⁡ open parenthesis θ close parenthesis or y equals A sin ⁡ open parenthesis θ close parenthesis")[$bold(italic(y)) = bold(italic(A)) bold(cos) ( bold(italic(θ)) ) " " " " " " "or" " " " " " " bold(italic(y)) = bold(italic(A)) bold(sin) ( bold(italic(θ)) )$] has amplitude #math.equation(block: false, alt: "vertical bar A vertical bar")[$| A |$].
+ The graph of #math.equation(block: true, alt: "y equals cos ⁡ open parenthesis B θ close parenthesis or y equals sin ⁡ open parenthesis B θ close parenthesis")[$bold(italic(y)) = bold(cos) ( bold(italic(B)) bold(italic(θ)) ) " " " " " " "or" " " " " " " bold(italic(y)) = bold(sin) ( bold(italic(B)) bold(italic(θ)) )$] has period #math.equation(block: false, alt: "the fraction 360 degrees over vertical bar B vertical bar")[$display(frac(360^(∘), | B |))$].
+ The graph of #math.equation(block: true, alt: "y equals k plus cos ⁡ open parenthesis θ close parenthesis or y equals k plus sin ⁡ open parenthesis θ close parenthesis")[$bold(italic(y)) = bold(italic(k)) + bold(cos) ( bold(italic(θ)) ) " " " " " " "or" " " " " " " bold(italic(y)) = bold(italic(k)) + bold(sin) ( bold(italic(θ)) )$] has midline #math.equation(block: false, alt: "y equals k")[$y = k$].
]


      
        Sketch a graph for each of the following functions.  Describe how each is different from the graph of #math.equation(block: false, alt: "y equals cos ⁡ θ")[$y = cos θ$].


        + #math.equation(block: false, alt: "h open parenthesis θ close parenthesis equals 2 cos ⁡ open parenthesis θ close parenthesis")[$h ( θ ) = 2 cos ( θ )$]
+ #math.equation(block: false, alt: "g open parenthesis θ close parenthesis equals cos ⁡ open parenthesis 2 θ close parenthesis")[$g ( θ ) = cos ( 2 θ )$]
+ #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals 2 plus cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = 2 + cos ( θ )$]


        
          + #figure(figph[graph of 2 cosine], alt: "graph of 2 cosine", caption: none)

The amplitude is 2.
+ #figure(figph[graph of cosine 2 theta], alt: "graph of cosine 2 theta", caption: none)

The period is #math.equation(block: false, alt: "180 degrees")[$180^(∘)$].
+ #figure(figph[graph of 2+cosine], alt: "graph of 2+cosine", caption: none)

The midline is #math.equation(block: false, alt: "y equals 2")[$y = 2$].


        
      
      The quantities #math.equation(block: false, alt: "A , B ,")[$A , B ,$] and #math.equation(block: false, alt: "k")[$k$] in the equations above are called #strong[parameters], and their values for a particular function give us information about its graph.


      #examplebox("Example 5")[][
State the period, midline, and amplitude of the graph of


          #math.equation(block: true, alt: "y equals minus 3 plus 4 sin ⁡ open parenthesis 3 θ close parenthesis")[$y = − 3 + 4 sin ( 3 θ )$]
          and graph the function.


        
        #solutionbox[
For this function, #math.equation(block: false, alt: "A equals 4 , B equals 3 ,")[$A = 4 , " " B = 3 ,$] and #math.equation(block: false, alt: "k equals minus 3")[$k = − 3$].  Its amplitude is #math.equation(block: false, alt: "4")[$4$], its period is #math.equation(block: false, alt: "the fraction 360 degrees over 3 equals 120 degrees")[$display(frac(360^(∘), 3)) = 120^(∘)$], and its midline is #math.equation(block: false, alt: "y equals minus 3")[$y = − 3$]. The graph is shown below.


          #figure(figph[graph of y= -3+4sin(3 theta)], alt: "graph of y= -3+4sin(3 theta)", caption: none)
]
]


      
        State the period, midline, and amplitude of the graph of


        #math.equation(block: true, alt: "y equals 1 minus 3 sin ⁡ open parenthesis 2 θ close parenthesis")[$y = 1 − 3 sin ( 2 θ )$]
        and graph the function.


        
          Amplitude #math.equation(block: false, alt: "3")[$3$], period #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], midline  #math.equation(block: false, alt: "y equals 1")[$y = 1$]


          #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
      
    
    #notebox("Algebra Refresher", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ Graph the function.
+ Give the coordinates of any intercepts, and any maximum or minimum values.


      + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals minus 6 plus the fraction 2 over 3 x")[$f ( x ) = − 6 + display(frac(2, 3)) x$]
+ #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 4 minus the fraction 3 over 2 x")[$g ( x ) = 4 − display(frac(3, 2)) x$]
+ #math.equation(block: false, alt: "p open parenthesis t close parenthesis equals t squared minus 4")[$p ( t ) = t^(2) − 4$]
+ #math.equation(block: false, alt: "q open parenthesis t close parenthesis equals 9 minus t squared")[$q ( t ) = 9 − t^(2)$]
+ #math.equation(block: false, alt: "F open parenthesis z close parenthesis equals 2 minus the square root of z")[$F ( z ) = 2 − sqrt(z)$]
+ #math.equation(block: false, alt: "G open parenthesis z close parenthesis equals the square root of 4 minus z")[$G ( z ) = sqrt(4 − z)$]


      #math.equation(block: true, alt: "bar")[$underline(#h(2em) #h(2em) #h(2em) #h(2em))$]


      Algebra Refresher Answers


      + + #figure(figph[graph], alt: "graph", caption: none)
+ #math.equation(block: false, alt: "open parenthesis 0 , minus 6 close parenthesis , open parenthesis 9 , 0 close parenthesis")[$( 0 , − 6 ) , " " ( 9 , 0 )$]
+ + #figure(figph[graph], alt: "graph", caption: none)
+ #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$], #math.equation(block: false, alt: "open parenthesis the fraction 8 over 3 , 0 close parenthesis")[$" " ( display(frac(8, 3)) , 0 )$]
+ + #figure(figph[parabola], alt: "parabola", caption: none)
+ #math.equation(block: false, alt: "open parenthesis 0 , minus 4 close parenthesis , open parenthesis minus 2 , 0 close parenthesis , open parenthesis 2 , 0 close parenthesis , Min: minus 4")[$( 0 , − 4 ) , " " ( − 2 , 0 ) , " " ( 2 , 0 ) , " " "Min:" " " − 4$]
+ + #figure(figph[parabola], alt: "parabola", caption: none)
+ #math.equation(block: false, alt: "open parenthesis 0 , 9 close parenthesis , open parenthesis minus 3 , 0 close parenthesis , open parenthesis 3 , 0 close parenthesis , Max: 9")[$( 0 , 9 ) , " " ( − 3 , 0 ) , " " ( 3 , 0 ) , " " "Max:" " " 9$]
+ + #figure(figph[graph], alt: "graph", caption: none)
+ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis , open parenthesis 4 , 0 close parenthesis , Max: 2")[$( 0 , 2 ) , " " ( 4 , 0 ) , " " "Max:" " " 2$]
+ + #figure(figph[graph], alt: "graph", caption: none)
+ #math.equation(block: false, alt: "open parenthesis 0 , 2 close parenthesis , open parenthesis 4 , 0 close parenthesis , Min: 0")[$( 0 , 2 ) , " " ( 4 , 0 ) , " " "Min:" " " 0$]
]


    
      
=== Section 4.2 Summary


      
        
==== Vocabulary


        - Input variable
- Output variable
- Periodic function
- Period
- Midline
- Amplitude
- Asymptote


      
      
        
==== Concepts


        + We use the notation #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] to indicate that #math.equation(block: false, alt: "y")[$y$] is a function of #math.equation(block: false, alt: "x")[$x$], that is, #math.equation(block: false, alt: "x")[$x$] is the input variable and #math.equation(block: false, alt: "y")[$y$] is the output variable.
+ Periodic functions are used to model phenomena that exhibit cyclical behavior.
+ The trigonometric ratios #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$] and #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis")[$cos ( θ )$] are functions of the angle #math.equation(block: false, alt: "θ")[$θ$].
+ The #strong[period] of the sine function is #math.equation(block: false, alt: "360 degrees")[$360^(∘)$]. Its #strong[midline] is the horizontal line #math.equation(block: false, alt: "y equals 0")[$y = 0$], and the #strong[amplitude] of the sine function is 1.
+ The graph of the cosine function has the same period, midline, and amplitude as the graph of the sine function. However, the locations of the intercepts and of the maximum and minimum values are different.
+ The tangent function has period #math.equation(block: false, alt: "180 degrees")[$180^(∘)$]. It is undefined at odd multiples of #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], and is increasing on each interval of its domain.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Amplitude, Period, and Midline]

+ The graph of #math.equation(block: true, alt: "y equals A cos ⁡ open parenthesis θ close parenthesis or y equals A sin ⁡ open parenthesis θ close parenthesis")[$bold(italic(y)) = bold(italic(A)) bold(cos) ( bold(italic(θ)) ) " " " " " " "or" " " " " " " bold(italic(y)) = bold(italic(A)) bold(sin) ( bold(italic(θ)) )$] has amplitude #math.equation(block: false, alt: "vertical bar A vertical bar")[$| A |$].
+ The graph of #math.equation(block: true, alt: "y equals cos ⁡ open parenthesis B θ close parenthesis or y equals sin ⁡ 9")[$bold(italic(y)) = bold(cos) ( bold(italic(B)) bold(italic(θ)) ) " " " " " " "or" " " " " " " bold(italic(y)) = bold(sin) 9$] has period #math.equation(block: false, alt: "the fraction 360 degrees over vertical bar B vertical bar")[$display(frac(360^(∘), | B |))$].
+ The graph of #math.equation(block: true, alt: "y equals k plus cos ⁡ open parenthesis θ close parenthesis or y equals k plus sin ⁡ open parenthesis θ close parenthesis")[$bold(italic(y)) = bold(italic(k)) + bold(cos) ( bold(italic(θ)) ) " " " " " " "or" " " " " " " bold(italic(y)) = bold(italic(k)) + bold(sin) ( bold(italic(θ)) )$] has midline #math.equation(block: false, alt: "y equals k")[$y = k$].
]


      
      
        
==== Study Questions


        + Use the figure to help you fill in the blanks.

#figure(figph[circle], alt: "circle", caption: none)

+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] to #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] to #math.equation(block: false, alt: "270 degrees")[$270^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "270 degrees")[$270^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ Use the figure to help you fill in the blanks.

#figure(figph[circle], alt: "circle", caption: none)

+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] to #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] to #math.equation(block: false, alt: "270 degrees")[$270^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ As #math.equation(block: false, alt: "θ")[$θ$] increases from #math.equation(block: false, alt: "270 degrees")[$270^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$] \_\_\_\_\_\_ from \_\_\_\_ to \_\_\_\_ .
+ List several ways in which the graph of #math.equation(block: false, alt: "y equals tan ⁡ θ")[$y = tan θ$] is different from the graphs of #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis θ close parenthesis")[$y = sin ( θ )$] and #math.equation(block: false, alt: "y equals cos ⁡ open parenthesis θ close parenthesis")[$y = cos ( θ )$].
+ State the period, midline, and amplitude of the graph of #math.equation(block: false, alt: "H equals 5 plus 0.2 cos ⁡ open parenthesis 3 α close parenthesis")[$H = 5 + 0.2 cos ( 3 α )$].


      
      
        
==== Skills


        + Sketch graphs of the sine and cosine functions    \#1-4, 9-10, 19-22
+ Find the coordinates of points on a sine or cosine graph    \#5-8, 37-42
+ Use function notation    \#11-18
+ Graph the tangent function    \#23-24
+ Write an equation for a sine or cosine graph     \#25-30, 49-66
+ Graph a sine or cosine function and state the period, midline, and amplitude    \#31-36, 43-48


      
    
    
      
=== Homework 4.2


      
        + Prepare a graph with the horizontal axis scaled from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] in multiples of #math.equation(block: false, alt: "45 degrees")[$45^(∘)$].
+ Sketch a graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] by plotting points for multiples of #math.equation(block: false, alt: "45 degrees")[$45^(∘)$].


        
          #figure(figph[sine graph], alt: "sine graph", caption: none)


        
      
      
        + Prepare a graph with the horizontal axis scaled from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] in multiples of #math.equation(block: false, alt: "45 degrees")[$45^(∘)$].
+ Sketch a graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$] by plotting points for multiples of #math.equation(block: false, alt: "45 degrees")[$45^(∘)$].


      
      
        + Prepare a graph with the horizontal axis scaled from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] in multiples of #math.equation(block: false, alt: "30 degrees")[$30^(∘)$].
+ Sketch a graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$] by plotting points for multiples of #math.equation(block: false, alt: "30 degrees")[$30^(∘)$].


        
          #figure(figph[cosine graph], alt: "cosine graph", caption: none)


        
      
      
        + Prepare a graph with the horizontal axis scaled from #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] to #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] in multiples of #math.equation(block: false, alt: "30 degrees")[$30^(∘)$].
+ Sketch a graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] by plotting points for multiples of #math.equation(block: false, alt: "30 degrees")[$30^(∘)$].


      
      For Problems 5–8, give the coordinates of each point on the graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] or #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$].


      
        #figure(figph[sine graph], alt: "sine graph", caption: none)


        
          + #math.equation(block: false, alt: "open parenthesis minus 225 degrees , the fraction 1 over the square root of 2 close parenthesis")[$( − 225^(∘) , display(frac(1, sqrt(2))) )$]
+ #math.equation(block: false, alt: "open parenthesis minus 135 degrees , the fraction minus 1 over the square root of 2 close parenthesis")[$( − 135^(∘) , display(frac(− 1, sqrt(2))) )$]
+ #math.equation(block: false, alt: "open parenthesis minus 90 degrees , minus 1 close parenthesis")[$( − 90^(∘) , − 1 )$]
+ #math.equation(block: false, alt: "open parenthesis 45 degrees , the fraction 1 over the square root of 2 close parenthesis")[$( 45^(∘) , display(frac(1, sqrt(2))) )$]
+ #math.equation(block: false, alt: "open parenthesis 180 degrees , 0 close parenthesis")[$( 180^(∘) , 0 )$]
+ #math.equation(block: false, alt: "open parenthesis 315 degrees , the fraction minus 1 over the square root of 2 close parenthesis")[$( 315^(∘) , display(frac(− 1, sqrt(2))) )$]


        
      
      
        #figure(figph[sine graph], alt: "sine graph", caption: none)


      
      
        #figure(figph[cosine graph], alt: "cosine graph", caption: none)


        
          + #math.equation(block: false, alt: "open parenthesis minus 240 degrees , the fraction minus 1 over 2 close parenthesis")[$( − 240^(∘) , display(frac(− 1, 2)) )$]
+ #math.equation(block: false, alt: "open parenthesis minus 210 degrees , the fraction minus the square root of 3 over 2 close parenthesis")[$( − 210^(∘) , display(frac(− sqrt(3), 2)) )$]
+ #math.equation(block: false, alt: "open parenthesis minus 60 degrees , the fraction minus 1 over 2 close parenthesis")[$( − 60^(∘) , display(frac(− 1, 2)) )$]
+ #math.equation(block: false, alt: "open parenthesis 30 degrees , the fraction the square root of 3 over 2 close parenthesis")[$( 30^(∘) , display(frac(sqrt(3), 2)) )$]
+ #math.equation(block: false, alt: "open parenthesis 120 degrees , the fraction minus 1 over 2 close parenthesis")[$( 120^(∘) , display(frac(− 1, 2)) )$]
+ #math.equation(block: false, alt: "open parenthesis 270 degrees , 0 close parenthesis")[$( 270^(∘) , 0 )$]


        
      
      
        #figure(figph[cosine graph], alt: "cosine graph", caption: none)


      
      
        Make a short table of values like the one shown, and sketch the function by hand. Be sure to label the #math.equation(block: false, alt: "x")[$x$]-axis and #math.equation(block: false, alt: "y")[$y$]-axis appropriately.


        #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


        + #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$]
+ #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = cos ( θ )$]


        
          + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]],
))

#figure(figph[cosine graph], alt: "cosine graph", caption: none)
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]],
))

#figure(figph[cosine graph], alt: "cosine graph", caption: none)


        
      
      
        One of these graphs is #math.equation(block: false, alt: "y equals A sin ⁡ open parenthesis k θ close parenthesis")[$y = A sin ( k θ )$], and the other is #math.equation(block: false, alt: "y equals A cos ⁡ open parenthesis k θ close parenthesis")[$y = A cos ( k θ )$]. Explain how you know which is which.


        #figure(figph[sinusoidal graphs], alt: "sinusoidal graphs", caption: none)


      
      For Problems 11–18, evaluate the expression for #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = sin ( θ )$] and #math.equation(block: false, alt: "g open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$g ( θ ) = cos ( θ )$].


      
        #math.equation(block: true, alt: "3 plus f open parenthesis 30 degrees close parenthesis")[$3 + f ( 30^(∘) )$]


        
          #math.equation(block: true, alt: "the fraction 7 over 2")[$display(frac(7, 2))$]


        
      
      
        #math.equation(block: true, alt: "3 f open parenthesis 30 degrees close parenthesis")[$3 f ( 30^(∘) )$]


      
      
        #math.equation(block: true, alt: "4 g open parenthesis 225 degrees close parenthesis minus 1")[$4 g ( 225^(∘) ) − 1$]


        
          #math.equation(block: true, alt: "minus 2 the square root of 2 minus 1")[$− 2 sqrt(2) − 1$]


        
      
      
        #math.equation(block: true, alt: "minus 4 plus 2 g open parenthesis 225 degrees close parenthesis")[$− 4 + 2 g ( 225^(∘) )$]


      
      
        #math.equation(block: false, alt: "minus 2 f open parenthesis 3 θ close parenthesis")[$− 2 f ( 3 θ )$], for #math.equation(block: false, alt: "θ equals 90 degrees")[$θ = 90^(∘)$]


        
          #math.equation(block: true, alt: "2")[$2$]


        
      
      
        #math.equation(block: false, alt: "6 f open parenthesis the fraction θ over 2 close parenthesis")[$6 f ( display(frac(θ, 2)) )$], for #math.equation(block: false, alt: "θ equals 90 degrees")[$θ = 90^(∘)$]


      
      
        #math.equation(block: false, alt: "8 minus 5 g open parenthesis the fraction θ over 3 close parenthesis")[$8 − 5 g ( display(frac(θ, 3)) )$], for #math.equation(block: false, alt: "θ equals 360 degrees")[$θ = 360^(∘)$]


        
          #math.equation(block: true, alt: "the fraction 21 over 2")[$display(frac(21, 2))$]


        
      
      
        #math.equation(block: false, alt: "1 minus 4 g open parenthesis 4 θ close parenthesis")[$1 − 4 g ( 4 θ )$], for #math.equation(block: false, alt: "θ equals 135 degrees")[$θ = 135^(∘)$]


      
      
        The graph shows your height as a function of angle as you ride the Ferris wheel. For each location #math.equation(block: false, alt: "A")[$A$]–#math.equation(block: false, alt: "E")[$E$] on the Ferris wheel, mark the corresponding point on the graph.


        #figure(figph[circle and graph], alt: "circle and graph", caption: none)


        
          #figure(figph[sine graph], alt: "sine graph", caption: none)


        
      
      
        The graph shows your height as a function of angle as you ride the Ferris wheel. For each location #math.equation(block: false, alt: "F")[$F$]–#math.equation(block: false, alt: "J")[$J$] on the graph, mark the corresponding point on the Ferris wheel.


        #figure(figph[circle and graph], alt: "circle and graph", caption: none)


      
      
        The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location #math.equation(block: false, alt: "K")[$K$]–#math.equation(block: false, alt: "O")[$O$] on the chain gear, mark the corresponding point on the graph.


        #figure(figph[circle and graph], alt: "circle and graph", caption: none)


        
          #figure(figph[graph], alt: "graph", caption: none)


        
      
      
        The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location  #math.equation(block: false, alt: "P")[$P$]–#math.equation(block: false, alt: "T")[$T$] on the graph, mark the corresponding point on the chain gear.


        #figure(figph[circle and graph], alt: "circle and graph", caption: none)


      
      
        + Fill in the table for values of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]. Round your answers to three decimal places.#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "81 degrees")[$81^(∘)$]], [#math.equation(block: false, alt: "82 degrees")[$82^(∘)$]], [#math.equation(block: false, alt: "83 degrees")[$83^(∘)$]], [#math.equation(block: false, alt: "84 degrees")[$84^(∘)$]], [#math.equation(block: false, alt: "85 degrees")[$85^(∘)$]], [#math.equation(block: false, alt: "86 degrees")[$86^(∘)$]], [#math.equation(block: false, alt: "87 degrees")[$87^(∘)$]], [#math.equation(block: false, alt: "88 degrees")[$88^(∘)$]], [#math.equation(block: false, alt: "89 degrees")[$89^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ What happens to #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] as #math.equation(block: false, alt: "θ")[$θ$] increases toward #math.equation(block: false, alt: "90 degrees")[$90^(∘)$]?
+ Fill in the table for values of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]. Round your answers to three decimal places.#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "99 degrees")[$99^(∘)$]], [#math.equation(block: false, alt: "98 degrees")[$98^(∘)$]], [#math.equation(block: false, alt: "97 degrees")[$97^(∘)$]], [#math.equation(block: false, alt: "96 degrees")[$96^(∘)$]], [#math.equation(block: false, alt: "95 degrees")[$95^(∘)$]], [#math.equation(block: false, alt: "94 degrees")[$94^(∘)$]], [#math.equation(block: false, alt: "93 degrees")[$93^(∘)$]], [#math.equation(block: false, alt: "92 degrees")[$92^(∘)$]], [#math.equation(block: false, alt: "91 degrees")[$91^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ What happens to #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] as #math.equation(block: false, alt: "θ")[$θ$] decreases toward #math.equation(block: false, alt: "90 degrees")[$90^(∘)$]?
+ What value does your calculator give for #math.equation(block: false, alt: "tan ⁡ open parenthesis 90 degrees close parenthesis")[$tan ( 90^(∘) )$]?  Why?


        
          + #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "81 degrees")[$81^(∘)$]], [#math.equation(block: false, alt: "82 degrees")[$82^(∘)$]], [#math.equation(block: false, alt: "83 degrees")[$83^(∘)$]], [#math.equation(block: false, alt: "84 degrees")[$84^(∘)$]], [#math.equation(block: false, alt: "85 degrees")[$85^(∘)$]], [#math.equation(block: false, alt: "86 degrees")[$86^(∘)$]], [#math.equation(block: false, alt: "87 degrees")[$87^(∘)$]], [#math.equation(block: false, alt: "88 degrees")[$88^(∘)$]], [#math.equation(block: false, alt: "89 degrees")[$89^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [#math.equation(block: false, alt: "6.314")[$6.314$]], [#math.equation(block: false, alt: "7.115")[$7.115$]], [#math.equation(block: false, alt: "8.144")[$8.144$]], [#math.equation(block: false, alt: "9.514")[$9.514$]], [#math.equation(block: false, alt: "11.43")[$11.43$]], [#math.equation(block: false, alt: "14.301")[$14.301$]], [#math.equation(block: false, alt: "19.081")[$19.081$]], [#math.equation(block: false, alt: "28.636")[$28.636$]], [#math.equation(block: false, alt: "57.29")[$57.29$]],
))
+ #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis approaches ∞")[$tan ( θ ) " " "approaches" " " upright(∞)$]
+ #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "99 degrees")[$99^(∘)$]], [#math.equation(block: false, alt: "98 degrees")[$98^(∘)$]], [#math.equation(block: false, alt: "97 degrees")[$97^(∘)$]], [#math.equation(block: false, alt: "96 degrees")[$96^(∘)$]], [#math.equation(block: false, alt: "95 degrees")[$95^(∘)$]], [#math.equation(block: false, alt: "94 degrees")[$94^(∘)$]], [#math.equation(block: false, alt: "93 degrees")[$93^(∘)$]], [#math.equation(block: false, alt: "92 degrees")[$92^(∘)$]], [#math.equation(block: false, alt: "91 degrees")[$91^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [#math.equation(block: false, alt: "minus 6.314")[$− 6.314$]], [#math.equation(block: false, alt: "minus 7.115")[$− 7.115$]], [#math.equation(block: false, alt: "minus 8.144")[$− 8.144$]], [#math.equation(block: false, alt: "minus 9.514")[$− 9.514$]], [#math.equation(block: false, alt: "minus 11.43")[$− 11.43$]], [#math.equation(block: false, alt: "minus 14.301")[$− 14.301$]], [#math.equation(block: false, alt: "minus 19.081")[$− 19.081$]], [#math.equation(block: false, alt: "minus 28.636")[$− 28.636$]], [#math.equation(block: false, alt: "minus 57.29")[$− 57.29$]],
))
+ #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis approaches minus ∞")[$tan ( θ ) " " "approaches" " " − upright(∞)$]
+ The calculator gives an error message because #math.equation(block: false, alt: "tan ⁡ open parenthesis 90 degrees close parenthesis")[$tan ( 90^(∘) )$] is undefined.


        
      
      
        + Fill in the table with exact values of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]. Then give decimal approximations to two places.#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "30 degrees")[$30^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "120 degrees")[$120^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "150 degrees")[$150^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] (exact)], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] (approx.)], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ Fill in the table with exact values of #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]. Then give decimal approximations to two places.#figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "210 degrees")[$210^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "240 degrees")[$240^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "300 degrees")[$300^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "330 degrees")[$330^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] (exact)], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$] (approx.)], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))
+ Plot the points from the tables and sketch a graph of #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals tan ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = tan ( θ )$].#figure(figph[tan grid], alt: "tan grid", caption: none)


      
      
        Write an equation for a sine function with amplitude 6.


        
          #math.equation(block: true, alt: "y equals 6 sin ⁡ open parenthesis θ close parenthesis")[$y = 6 sin ( θ )$]


        
      
      
        Write an equation for a cosine function with amplitude #math.equation(block: false, alt: "the fraction 1 over 2")[$display(frac(1, 2))$].


      
      
        Write an equation for a cosine function with midline #math.equation(block: false, alt: "minus 5")[$− 5$].


        
          #math.equation(block: true, alt: "y equals cos ⁡ open parenthesis θ close parenthesis minus 5")[$y = cos ( θ ) − 5$]


        
      
      
        Write an equation for a sine function with midline 2.


      
      
        Write an equation for a sine function with period #math.equation(block: false, alt: "90 degrees")[$90^(∘)$].


        
          #math.equation(block: true, alt: "y equals sin ⁡ open parenthesis 4 θ close parenthesis")[$y = sin ( 4 θ )$]


        
      
      
        Write an equation for a cosine function with period #math.equation(block: false, alt: "720 degrees")[$720^(∘)$].


      
      For Problems 31–36,


      + Graph the function.
+ State the amplitude, period and midline of the function.


      
        #math.equation(block: true, alt: "y equals 3 cos ⁡ open parenthesis θ close parenthesis")[$y = 3 cos ( θ )$]


        #figure(figph[grid], alt: "grid", caption: none)


        
          #figure(figph[graph of y = 3 cos theta], alt: "graph of y = 3 cos theta", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals 4 sin ⁡ open parenthesis θ close parenthesis")[$y = 4 sin ( θ )$]


        #figure(figph[grid], alt: "grid", caption: none)


      
      
        #math.equation(block: true, alt: "y equals 3 plus sin ⁡ open parenthesis θ close parenthesis")[$y = 3 + sin ( θ )$]


        #figure(figph[grid], alt: "grid", caption: none)


        
          #figure(figph[graph of 3 + sin theta], alt: "graph of 3 + sin theta", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals minus 2 plus cos ⁡ open parenthesis θ close parenthesis")[$y = − 2 + cos ( θ )$]


        #figure(figph[grid], alt: "grid", caption: none)


      
      
        #math.equation(block: true, alt: "y equals cos ⁡ open parenthesis 3 θ close parenthesis")[$y = cos ( 3 θ )$]


        #figure(figph[grid], alt: "grid", caption: none)


        
          #figure(figph[graph of cos 3 theta], alt: "graph of cos 3 theta", caption: none)


        
      
      
        #math.equation(block: true, alt: "y equals sin ⁡ open parenthesis 2 θ close parenthesis")[$y = sin ( 2 θ )$]


        #figure(figph[grid], alt: "grid", caption: none)


      
      For Problems 37–42, give the coordinates of the points on the graph.


      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals minus 3 cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = − 3 cos ( θ )$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "A open parenthesis 0 degrees , minus 3 close parenthesis")[$A ( 0^(∘) , − 3 )$], #math.equation(block: true, alt: "B open parenthesis 135 degrees , the fraction 3 over the square root of 2 close parenthesis")[$" " B ( 135^(∘) , display(frac(3, sqrt(2))) )$], #math.equation(block: true, alt: "C open parenthesis 300 degrees , the fraction minus 3 over 2 close parenthesis")[$" " C ( 300^(∘) , display(frac(− 3, 2)) )$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals minus 4 sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = − 4 sin ( θ )$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals sin ⁡ open parenthesis 4 θ close parenthesis")[$f ( θ ) = sin ( 4 θ )$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "P open parenthesis 112.5 degrees , 1 close parenthesis")[$P ( 112.5^(∘) , 1 )$], #math.equation(block: true, alt: "Q open parenthesis 180 degrees , 0 close parenthesis")[$" " Q ( 180^(∘) , 0 )$], #math.equation(block: true, alt: "R open parenthesis 337.5 degrees , minus 1 close parenthesis")[$" " R ( 337.5^(∘) , − 1 )$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals cos ⁡ open parenthesis 3 θ close parenthesis")[$f ( θ ) = cos ( 3 θ )$]


        #figure(figph[triangle], alt: "triangle", caption: none)


      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals minus 3 plus cos ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = − 3 + cos ( θ )$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          #math.equation(block: true, alt: "X open parenthesis 45 degrees , minus 3 plus the fraction 1 over the square root of 2 close parenthesis")[$X ( 45^(∘) , − 3 + display(frac(1, sqrt(2))) )$], #math.equation(block: true, alt: "Y open parenthesis 90 degrees , minus 3 close parenthesis")[$" " Y ( 90^(∘) , − 3 )$], #math.equation(block: true, alt: "Z open parenthesis 300 degrees , minus 2 close parenthesis")[$" " Z ( 300^(∘) , − 2 )$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals 1 plus sin ⁡ open parenthesis θ close parenthesis")[$f ( θ ) = 1 + sin ( θ )$]


        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      For Problems 43–48, graph the function using technology. State the amplitude, period, and midline.


      
        #math.equation(block: true, alt: "y equals 3 plus 4 cos ⁡ open parenthesis θ close parenthesis")[$y = 3 + 4 cos ( θ )$]


        
          amp#math.equation(block: false, alt: "equals 4")[$= 4$], period #math.equation(block: false, alt: "equals 360 degrees")[$= 360^(∘)$], midline: #math.equation(block: false, alt: "y equals 3")[$y = 3$]


        
      
      
        #math.equation(block: true, alt: "y equals minus 4 plus 3 sin ⁡ open parenthesis θ close parenthesis")[$y = − 4 + 3 sin ( θ )$]


      
      
        #math.equation(block: true, alt: "y equals 5 sin ⁡ open parenthesis 2 θ close parenthesis")[$y = 5 sin ( 2 θ )$]


        
          amp#math.equation(block: false, alt: "equals 5")[$= 5$], period #math.equation(block: false, alt: "equals 180 degrees")[$= 180^(∘)$], midline: #math.equation(block: false, alt: "y equals 0")[$y = 0$]


        
      
      
        #math.equation(block: true, alt: "y equals 6 cos ⁡ open parenthesis 4 θ close parenthesis")[$y = 6 cos ( 4 θ )$]


      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals minus 4 plus 3 sin ⁡ open parenthesis 3 θ close parenthesis")[$f ( θ ) = − 4 + 3 sin ( 3 θ )$]


        
          amp#math.equation(block: false, alt: "equals 3")[$= 3$], period #math.equation(block: false, alt: "equals 120 degrees")[$= 120^(∘)$], midline: #math.equation(block: false, alt: "y equals minus 4")[$y = − 4$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis θ close parenthesis equals 2 plus 4 cos ⁡ open parenthesis 3 θ close parenthesis")[$f ( θ ) = 2 + 4 cos ( 3 θ )$]


      
      For Problems 49–56,


      + State the amplitude, period, and midline for the graph.
+ Write an equation for the graph using sine or cosine.


      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amp #math.equation(block: false, alt: "equals 1")[$= 1$], period #math.equation(block: false, alt: "equals 90 degrees")[$= 90^(∘)$], midline: #math.equation(block: false, alt: "y equals 0")[$y = 0$]
+ #math.equation(block: false, alt: "y equals sin ⁡ open parenthesis 4 θ close parenthesis")[$y = sin ( 4 θ )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amp #math.equation(block: false, alt: "equals 1")[$= 1$], period #math.equation(block: false, alt: "equals 360 degrees")[$= 360^(∘)$], midline: #math.equation(block: false, alt: "y equals 3")[$y = 3$]
+ #math.equation(block: false, alt: "y equals 3 plus cos ⁡ open parenthesis θ close parenthesis")[$y = 3 + cos ( θ )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amp #math.equation(block: false, alt: "equals 4")[$= 4$], period #math.equation(block: false, alt: "equals 360 degrees")[$= 360^(∘)$], midline: #math.equation(block: false, alt: "y equals minus 2")[$y = − 2$]
+ #math.equation(block: false, alt: "y equals minus 2 plus 4 sin ⁡ open parenthesis θ close parenthesis")[$y = − 2 + 4 sin ( θ )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


        
          + amp #math.equation(block: false, alt: "equals 2")[$= 2$], period #math.equation(block: false, alt: "equals 120 degrees")[$= 120^(∘)$], midline: #math.equation(block: false, alt: "y equals 2")[$y = 2$]
+ #math.equation(block: false, alt: "y equals 2 plus 2 cos ⁡ open parenthesis 3 θ close parenthesis")[$y = 2 + 2 cos ( 3 θ )$]


        
      
      
        #figure(figph[sinusoidal graph], alt: "sinusoidal graph", caption: none)


      
      For Problems 57–62, write the equation of a sine or cosine function with the given properties.


      
        Midline #math.equation(block: false, alt: "y equals minus 4")[$y = − 4$], amplitude #math.equation(block: false, alt: "6")[$6$], period #math.equation(block: false, alt: "120 degrees")[$120^(∘)$]


        
          #math.equation(block: false, alt: "y equals minus 4 plus 6 sin ⁡ open parenthesis 3 θ close parenthesis")[$y = − 4 + 6 sin ( 3 θ )$] (Answers vary)


        
      
      
        Midline #math.equation(block: false, alt: "y equals 5")[$y = 5$], amplitude #math.equation(block: false, alt: "the fraction 3 over 2")[$display(frac(3, 2))$], period #math.equation(block: false, alt: "180 degrees")[$180^(∘)$]


      
      
        Maximum points at #math.equation(block: false, alt: "open parenthesis 0 degrees , 5 close parenthesis")[$( 0^(∘) , 5 )$] and #math.equation(block: false, alt: "open parenthesis 360 degrees , 5 close parenthesis")[$( 360^(∘) , 5 )$], minimum point at #math.equation(block: false, alt: "open parenthesis 180 degrees , 1 close parenthesis")[$( 180^(∘) , 1 )$]


        
          #math.equation(block: false, alt: "y equals 3 plus 2 cos ⁡ open parenthesis θ close parenthesis")[$y = 3 + 2 cos ( θ )$] (Answers vary)


        
      
      
        Maximum point at #math.equation(block: false, alt: "open parenthesis 90 degrees , 1 close parenthesis")[$( 90^(∘) , 1 )$], minimum point at #math.equation(block: false, alt: "open parenthesis 270 degrees , minus 3 close parenthesis")[$( 270^(∘) , − 3 )$]


      
      
        Horizontal intercepts at #math.equation(block: false, alt: "45 degrees")[$45^(∘)$] and #math.equation(block: false, alt: "135 degrees")[$135^(∘)$], vertical intercept at #math.equation(block: false, alt: "open parenthesis 0 degrees , 12 close parenthesis")[$( 0^(∘) , 12 )$]


        
          #math.equation(block: false, alt: "y equals 12 cos ⁡ open parenthesis 2 θ close parenthesis")[$y = 12 cos ( 2 θ )$] (Answers vary)


        
      
      
        Horizontal intercepts at #math.equation(block: false, alt: "30 degrees")[$30^(∘)$] and #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], vertical intercept at #math.equation(block: false, alt: "open parenthesis 0 degrees , minus 8 close parenthesis")[$( 0^(∘) , − 8 )$]


      
      For Problems 63–66, the table describes a sine or cosine function. Find an equation for the function.


      
        #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "5.56")[$5.56$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 1.54")[$− 1.54$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 1.54")[$− 1.54$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "5.54")[$5.54$]], [#math.equation(block: false, alt: "7")[$7$]],
))


        
          #math.equation(block: true, alt: "y equals 2 plus 5 cos ⁡ open parenthesis θ close parenthesis")[$y = 2 + 5 cos ( θ )$]


        
      
      
        #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "3.12")[$3.12$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3.12")[$3.12$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 1.12")[$− 1.12$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1.12")[$− 1.12$]], [#math.equation(block: false, alt: "1")[$1$]],
))


      
      
        #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 2.83")[$− 2.83$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 2.83")[$− 2.83$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2.83")[$2.83$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "2.83")[$2.83$]], [#math.equation(block: false, alt: "0")[$0$]],
))


        
          #math.equation(block: true, alt: "y equals minus 4 sin ⁡ open parenthesis θ close parenthesis")[$y = − 4 sin ( θ )$]


        
      
      
        #figure(table(
  columns: 10, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#math.equation(block: false, alt: "0 degrees")[$0^(∘)$]], [#math.equation(block: false, alt: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "90 degrees")[$90^(∘)$]], [#math.equation(block: false, alt: "135 degrees")[$135^(∘)$]], [#math.equation(block: false, alt: "180 degrees")[$180^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "270 degrees")[$270^(∘)$]], [#math.equation(block: false, alt: "315 degrees")[$315^(∘)$]], [#math.equation(block: false, alt: "360 degrees")[$360^(∘)$]]),
  [#math.equation(block: false, alt: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]], [#math.equation(block: false, alt: "minus 6.36")[$− 6.36$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "6.36")[$6.36$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "6.36")[$6.36$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 6.36")[$− 6.36$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]],
))