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== 4.4#h(0.6em)Chapter 4 Summary and Review


    
      
=== Key Concepts


      + We can use angles to describe rotation. Positive angles indicate rotation in the counter-clockwise direction; negative angles describe clockwise rotation.
+ We define the trigonometric ratios of any angle by placing the angle in standard position and choosing a point  on the terminal side, with #math.equation(block: false, alt: "r equals the square root of x squared plus y squared")[$r = sqrt(x^(2) + y^(2))$].
        #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[The Trigonometric Ratios]

If #math.equation(block: false, alt: "θ")[$θ$] is an angle in standard position, and #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] is a point on its terminal side, with  #math.equation(block: false, alt: "r equals the square root of x squared plus y squared")[$r = sqrt(x^(2) + y^(2))$],  then

#math.equation(block: true, alt: "sin ⁡ θ equals the fraction y over r cos ⁡ θ equals the fraction x over r tan ⁡ θ equals the fraction y over x")[$sin θ = display(frac(y, r)) " " " " " " " " " " " " " " " " " " cos θ = display(frac(x, r)) " " " " " " " " " " " " " " " " " " tan θ = display(frac(y, x))$]
]
+ To construct a reference triangle for an angle :+ Choose a point #math.equation(block: false, alt: "P")[$P$] on the terminal side.
+ Draw a line from point #math.equation(block: false, alt: "P")[$P$] perpendicular to the #math.equation(block: false, alt: "x")[$x$]-axis.
+ The #strong[reference angle] for #math.equation(block: false, alt: "θ")[$θ$] is the positive acute angle formed between the terminal side of #math.equation(block: false, alt: "θ")[$θ$] and the #math.equation(block: false, alt: "x")[$x$]-axis. 
        #figure(figph[Reference angles for all quadrants], alt: "Reference angles for all quadrants", caption: none)
+ The trigonometric ratios of any angle are equal to the ratios of its reference angle, except for sign. The sign of the ratio is determined by the quadrant.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[To find an angle #math.equation(block: false, alt: "θ")[$θ$] with a given reference angle #math.equation(block: false, alt: "θ tilde")[$tilde(θ)$]:]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([Quadrant I:    #math.equation(block: false, alt: "θ equals θ tilde")[$" " " " " " " " " " " " θ = tilde(θ)$]], [$#hide($0000$)$], [Quadrant II:  #math.equation(block: false, alt: "θ equals 180 degrees minus θ tilde")[$" " " " " " " " θ = 180^(∘) − tilde(θ)$]]),
  [Quadrant III:  #math.equation(block: false, alt: "θ equals 180 degrees plus θ tilde")[$" " " " " " " " " " θ = 180^(∘) + tilde(θ)$]], [$#hide($0000$)$], [Quadrant IV:  #math.equation(block: false, alt: "θ equals 360 degrees minus θ tilde")[$" " " " " " " " θ = 360^(∘) − tilde(θ)$]],
))
]


        #figure(figph[angles], alt: "angles", caption: none)
+ There are always two angles between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] (except for the quadrantal angles) with a given trigonometric ratio.
+ #strong[Coterminal angles] have equal trigonometric ratios.
+ To solve an equation of the form #math.equation(block: false, alt: "sin ⁡ θ equals k")[$sin θ = k$], or #math.equation(block: false, alt: "cos ⁡ θ equals k")[$cos θ = k$], or #math.equation(block: false, alt: "tan ⁡ θ equals k")[$tan θ = k$], we can use the appropriate inverse trig key on a calculator to find one solution (or a coterminal angle.)  We use reference angles to find a second solution between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$].
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Angles in a Unit Circle]

Let #math.equation(block: false, alt: "P")[$P$] be a point on a unit circle determined by the terminal side of an angle #math.equation(block: false, alt: "θ")[$θ$] in standard position. Then the coordinates #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$]  of #math.equation(block: false, alt: "P")[$P$] are given by

#math.equation(block: true, alt: "x equals cos ⁡ θ , y equals sin ⁡ θ")[$x = cos θ , " " " " " " " " " " " " y = sin θ$]
]
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Coordinates]

If point #math.equation(block: false, alt: "P")[$P$] is located at a distance #math.equation(block: false, alt: "r")[$r$] from the origin in the direction specified by angle #math.equation(block: false, alt: "θ")[$θ$] in standard position, then the coordinates of #math.equation(block: false, alt: "P")[$P$] are

#math.equation(block: true, alt: "x equals r cos ⁡ θ and y equals r sin ⁡ θ")[$x = r cos θ " " " " " " " " "and" " " " " " " " " y = r sin θ$]
]
+ Navigational directions for ships and planes are sometimes given as #strong[bearings], which are angles measured clockwise from north.
+ Periodic functions are used to model phenomena that exhibit cyclical behavior.
+ The trigonometric ratios #math.equation(block: false, alt: "sin ⁡ θ")[$sin θ$] and #math.equation(block: false, alt: "cos ⁡ θ")[$cos θ$] 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.
+ 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.
+ 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[Angle of Inclination]

The #strong[angle of inclination] of a line is the angle #math.equation(block: false, alt: "α")[$α$] measured in the positive direction from the positive #math.equation(block: false, alt: "x")[$x$]-axis to the line. If the slope of the line is #math.equation(block: false, alt: "m")[$m$], then

#math.equation(block: true, alt: "tan ⁡ α equals m")[$tan α = m$]where  #math.equation(block: false, alt: "0 degrees less than or equal to α less than or equal to 180 degrees")[$0^(∘) ≤ α ≤ 180^(∘)$].
]
+ #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 ⁡ θ or y equals A sin ⁡ θ")[$y = A cos θ " " " " "or" " " " " y = A sin θ$]  has amplitude #math.equation(block: false, alt: "vertical bar A vertical bar")[$| A |$].
+ The graph of #math.equation(block: true, alt: "y equals cos ⁡ B θ or y equals sin ⁡ B θ")[$y = cos B θ " " " " "or" " " " " y = sin B θ$]  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 ⁡ θ or y equals k plus sin ⁡ θ")[$y = k + cos θ " " " " "or" " " " " y = k + sin θ$]  has midline #math.equation(block: false, alt: "y equals k")[$y = k$].
]
+ The graph of #math.equation(block: false, alt: "y equals k plus A sin ⁡ B θ")[$y = k + A sin B θ$] has amplitude #math.equation(block: false, alt: "A")[$A$], period #math.equation(block: false, alt: "the fraction 360 degrees over B")[$display(frac(360^(∘), B))$], and midline #math.equation(block: false, alt: "y equals k")[$y = k$]. The same is true for the graph of #math.equation(block: false, alt: "y equals k plus A cos ⁡ B θ")[$y = k + A cos B θ$].
+ Functions that have graphs shaped like sines or cosines are called #strong[sinusoidal].
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Periodic Function]

The function #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$] is #strong[periodic] if there is a smallest value of #math.equation(block: false, alt: "p")[$p$] such that

#math.equation(block: true, alt: "f open parenthesis x plus p close parenthesis equals f open parenthesis x close parenthesis")[$f ( x + p ) = f ( x )$]for all #math.equation(block: false, alt: "x")[$x$]. The constant #math.equation(block: false, alt: "p")[$p$]  is called the #strong[period] of the function.
]


    
    
      
=== Chapter 4 Review Problems


      
        The London Eye, the world's largest Ferris wheel, completes one revolution every 30 minutes.  By how many degrees will it rotate in 1 minute?


        
          #math.equation(block: true, alt: "12 degrees")[$12^(∘)$]


        
      
      
        The London Eye in Problem 1 has 32 cabins evenly spaced along the wheel.  If the cabins are numbered consecutively from 1 to 32, what is the angular separation between cabins number 1 and number 15?


      
      For Problems 3–4,find two angles, one positive and one negative, that are coterminal with the given angle.


      
        + #math.equation(block: false, alt: "510 degrees")[$510^(∘)$]
+ #math.equation(block: false, alt: "600 degrees")[$600^(∘)$]
+ #math.equation(block: false, alt: "minus 200 degrees")[$− 200^(∘)$]
+ #math.equation(block: false, alt: "minus 700 degrees")[$− 700^(∘)$]


        
          + #math.equation(block: false, alt: "150 degrees , minus 210 degrees")[$150^(∘) , " " − 210^(∘)$]
+ #math.equation(block: false, alt: "240 degrees , minus 120 degrees")[$240^(∘) , " " − 120^(∘)$]
+ #math.equation(block: false, alt: "160 degrees , minus 560 degrees")[$160^(∘) , " " − 560^(∘)$]
+ #math.equation(block: false, alt: "20 degrees , minus 340 degrees")[$20^(∘) , " " − 340^(∘)$]


        
      
      
        + #math.equation(block: false, alt: "minus 380 degrees")[$− 380^(∘)$]
+ #math.equation(block: false, alt: "minus 423 degrees")[$− 423^(∘)$]
+ #math.equation(block: false, alt: "187 degrees")[$187^(∘)$]
+ #math.equation(block: false, alt: "1000 degrees")[$1000^(∘)$]


      
      For the angles in Problems 5–6, state the corresponding quadrant and reference angle.  Give three other angles with the same reference angle, one for each of the other three quadrants.  Sketch all four angles.


      
        + #math.equation(block: false, alt: "minus 300 degrees")[$− 300^(∘)$]
+ #math.equation(block: false, alt: "minus 25 degrees")[$− 25^(∘)$]
+ #math.equation(block: false, alt: "100 degrees")[$100^(∘)$]
+ #math.equation(block: false, alt: "250 degrees")[$250^(∘)$]


        
          + #math.equation(block: false, alt: "I , 60 degrees ; 120 degrees , 240 degrees , 300 degrees")[$I , " " 60^(∘) ; " " 120^(∘) , " " 240^(∘) , " " 300^(∘)$]
+ #math.equation(block: false, alt: "I V , 25 degrees ; 155 degrees , 205 degrees , 335 degrees")[$I V , " " 25^(∘) ; " " 155^(∘) , " " 205^(∘) , " " 335^(∘)$]
+ #math.equation(block: false, alt: "I I , 80 degrees ; 80 degrees , 260 degrees , 280 degrees")[$I I , " " 80^(∘) ; " " 80^(∘) , " " 260^(∘) , " " 280^(∘)$]
+ #math.equation(block: false, alt: "I I I , 70 degrees ; 70 degrees , 110 degrees , 290 degrees")[$I I I , " " 70^(∘) ; " " 70^(∘) , " " 110^(∘) , " " 290^(∘)$]


        
      
      
        + #math.equation(block: false, alt: "430 degrees")[$430^(∘)$]
+ #math.equation(block: false, alt: "minus 590 degrees")[$− 590^(∘)$]
+ #math.equation(block: false, alt: "minus 95 degrees")[$− 95^(∘)$]
+ #math.equation(block: false, alt: "1050 degrees")[$1050^(∘)$]


      
      
        Let #math.equation(block: false, alt: "θ tilde equals f open parenthesis θ close parenthesis")[$tilde(θ) = f ( θ )$] be the function that gives the reference angle of #math.equation(block: false, alt: "θ")[$θ$].  For example, #math.equation(block: false, alt: "f open parenthesis 110 degrees close parenthesis equals 70 degrees")[$f ( 110^(∘) ) = 70^(∘)$] because the reference angle for #math.equation(block: false, alt: "110 degrees")[$110^(∘)$] is #math.equation(block: false, alt: "70 degrees")[$70^(∘)$].


        + Fill in the table of values.#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: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$],
))
+ Choose appropriate scales for the axes and graph the function for #math.equation(block: false, alt: "minus 360 degrees less than or equal to θ less than or equal to 360 degrees")[$− 360^(∘) ≤ θ ≤ 360^(∘)$].#figure(figph[grid], alt: "grid", caption: none)


        
          + #figure(table(
  columns: 13, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "θ")[$θ$]], [#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: "f open parenthesis θ close parenthesis")[$f ( θ )$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "90")[$90$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "90")[$90$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "0")[$0$]],
))
+ #figure(figph[graph of referance angle vs angle], alt: "graph of referance angle vs angle", caption: none)


        
      
      
        Let #math.equation(block: false, alt: "θ tilde equals f open parenthesis θ close parenthesis")[$tilde(θ) = f ( θ )$] be the function that gives the reference angle of #math.equation(block: false, alt: "θ")[$θ$].  (See Problem 7.)  Is #math.equation(block: false, alt: "f")[$f$] a periodic function?  If so, give its period, midline, and amplitude.  If not, explain why not.


      
      For Problems 9–20, solve the equation exactly 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: true, alt: "sin ⁡ θ equals the fraction minus 1 over 2")[$sin θ = display(frac(− 1, 2))$]


        
          #math.equation(block: true, alt: "210 degrees , 330 degrees")[$210^(∘) , " " 330^(∘)$]


        
      
      
        #math.equation(block: true, alt: "cos ⁡ θ equals the fraction minus 1 over the square root of 2")[$cos θ = display(frac(− 1, sqrt(2)))$]


      
      
        #math.equation(block: true, alt: "2 cos ⁡ θ plus 1 equals 0")[$2 cos θ + 1 = 0$]


        
          #math.equation(block: true, alt: "120 degrees , 240 degrees")[$120^(∘) , " " 240^(∘)$]


        
      
      
        #math.equation(block: true, alt: "5 sin ⁡ θ plus 5 equals 0")[$5 sin θ + 5 = 0$]


      
      
        #math.equation(block: true, alt: "tan ⁡ θ minus 1 equals 0")[$tan θ − 1 = 0$]


        
          #math.equation(block: true, alt: "45 degrees , 225 degrees")[$45^(∘) , " " 225^(∘)$]


        
      
      
        #math.equation(block: true, alt: "the square root of 3 plus 3 tan ⁡ θ equals 0")[$sqrt(3) + 3 tan θ = 0$]


      
      
        #math.equation(block: true, alt: "cos ⁡ θ equals cos ⁡ open parenthesis minus 23 degrees close parenthesis")[$cos θ = cos ( − 23^(∘) )$]


        
          #math.equation(block: true, alt: "23 degrees , 337 degrees")[$23^(∘) , " " 337^(∘)$]


        
      
      
        #math.equation(block: true, alt: "sin ⁡ θ equals sin ⁡ open parenthesis 370 degrees close parenthesis")[$sin θ = sin ( 370^(∘) )$]


      
      
        #math.equation(block: true, alt: "tan ⁡ θ equals tan ⁡ 432 degrees")[$tan θ = tan 432^(∘)$]


        
          #math.equation(block: true, alt: "72 degrees , 252 degrees")[$72^(∘) , " " 252^(∘)$]


        
      
      
        #math.equation(block: true, alt: "tan ⁡ θ equals tan ⁡ open parenthesis minus 6 degrees close parenthesis")[$tan θ = tan ( − 6^(∘) )$]


      
      
        #math.equation(block: true, alt: "sin ⁡ θ plus sin ⁡ 83 degrees equals 0")[$sin θ + sin 83^(∘) = 0$]


        
          #math.equation(block: true, alt: "163 degrees , 277 degrees")[$163^(∘) , " " 277^(∘)$]


        
      
      
        #math.equation(block: true, alt: "cos ⁡ θ plus cos ⁡ 429 degrees equals 0")[$cos θ + cos 429^(∘) = 0$]


      
      For Problems 21–26, solve the equation for #math.equation(block: false, alt: "0 degrees less than or equal to θ less than or equal to 360 degrees")[$0^(∘) ≤ θ ≤ 360^(∘)$]. Round your answers to two decimal places.


      
        #math.equation(block: true, alt: "3 sin ⁡ θ plus 2 equals 0")[$3 sin θ + 2 = 0$]


        
          #math.equation(block: true, alt: "221.81 degrees , 318.19 degrees")[$221.81^(∘) , " " 318.19^(∘)$]


        
      
      
        #math.equation(block: true, alt: "5 cos ⁡ θ plus 4 equals 0")[$5 cos θ + 4 = 0$]


      
      
        #math.equation(block: true, alt: "the fraction 2 over 3 tan ⁡ θ plus 1 equals 0")[$display(frac(2, 3)) tan θ + 1 = 0$]


        
          #math.equation(block: true, alt: "123.69 degrees , 303.69 degrees")[$123.69^(∘) , " " 303.69^(∘)$]


        
      
      
        #math.equation(block: true, alt: "minus 4 tan ⁡ θ plus 12 equals 0")[$− 4 tan θ + 12 = 0$]


      
      
        #math.equation(block: true, alt: "4 equals 8 cos ⁡ θ plus 9")[$4 = 8 cos θ + 9$]


        
          #math.equation(block: true, alt: "128.68 degrees , 231.32 degrees")[$128.68^(∘) , " " 231.32^(∘)$]


        
      
      
        #math.equation(block: true, alt: "minus 3 equals 6 sin ⁡ θ minus 5")[$− 3 = 6 sin θ − 5$]


      
      For Problems 27–32, find the coordinates of the point where the terminal side of angle #math.equation(block: false, alt: "θ")[$θ$] in standard position intersects the circle of radius #math.equation(block: false, alt: "r")[$r$] centered at the origin.  Round your answers to two decimal places.


      
        #math.equation(block: true, alt: "θ equals 193 degrees , r equals 10")[$θ = 193^(∘) , " " r = 10$]


        
          #math.equation(block: true, alt: "open parenthesis minus 9.74 , minus 2.25 close parenthesis")[$( − 9.74 , − 2.25 )$]


        
      
      
        #math.equation(block: true, alt: "θ equals minus 12 degrees , r equals 20")[$θ = − 12^(∘) , " " r = 20$]


      
      
        #math.equation(block: true, alt: "θ equals 92 degrees , r equals 8")[$θ = 92^(∘) , " " r = 8$]


        
          #math.equation(block: true, alt: "open parenthesis minus 0.28 , 8.00 close parenthesis")[$( − 0.28 , 8.00 )$]


        
      
      
        #math.equation(block: true, alt: "θ equals 403 degrees , r equals 6")[$θ = 403^(∘) , " " r = 6$]


      
      
        #math.equation(block: true, alt: "θ equals minus 341 degrees , r equals 3")[$θ = − 341^(∘) , " " r = 3$]


        
          #math.equation(block: true, alt: "open parenthesis 2.84 , 0.98 close parenthesis")[$( 2.84 , 0.98 )$]


        
      
      
        #math.equation(block: true, alt: "θ equals minus 107 degrees , r equals 20")[$θ = − 107^(∘) , " " r = 20$]


      
      
        If you follow a bearing of #math.equation(block: false, alt: "190 degrees")[$190^(∘)$] for 10 miles, how far south and how far west are you from your starting point?  Round to two decimal places.


        
          south: 1.74 mi, west: 9.85 mi


        
      
      
        A child releases a balloon, which then follows a bearing of #math.equation(block: false, alt: "84 degrees")[$84^(∘)$] for 150 meters.  How far east and how far north is the balloon from where it was released?  Round to the nearest meter.


      
      For Problems 35–38, write the equation of a sine or cosine function with the given properties and sketch the graph, including at least one period.


      
        Amplitude 7, midline #math.equation(block: false, alt: "y equals 4")[$y = 4$], period 2, #math.equation(block: false, alt: "y")[$y$]-intercept 4


        
          #math.equation(block: true, alt: "y equals 4 plus 7 sin ⁡ open parenthesis 180 θ close parenthesis")[$y = 4 + 7 sin ( 180 θ )$]


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


        
      
      
        Amplitude 100, midline #math.equation(block: false, alt: "y equals 50")[$y = 50$], period 12, #math.equation(block: false, alt: "y")[$y$]-intercept 100


      
      
        Maximum point at #math.equation(block: false, alt: "open parenthesis 90 degrees , 24 close parenthesis")[$( 90^(∘) , 24 )$] and minimum point at #math.equation(block: false, alt: "open parenthesis 270 degrees , 10 close parenthesis")[$( 270^(∘) , 10 )$]


        
          #math.equation(block: true, alt: "y equals 17 plus 7 sin ⁡ θ")[$y = 17 + 7 sin θ$]


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


        
      
      
        Horizontal intercept at #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")[$( 360^(∘) , 5 )$]


      
      For Problems 39–42, evaluate the expression for #math.equation(block: false, alt: "f open parenthesis θ close parenthesis equals sin ⁡ θ")[$f ( θ ) = sin θ$]  and #math.equation(block: false, alt: "g open parenthesis θ close parenthesis equals cos ⁡ θ")[$g ( θ ) = cos θ$].


      
        #math.equation(block: false, alt: "2 f open parenthesis θ close parenthesis g open parenthesis θ close parenthesis")[$2 f ( θ ) g ( θ )$] for #math.equation(block: false, alt: "θ equals 30 degrees")[$θ = 30^(∘)$]


        
          #math.equation(block: true, alt: "the fraction the square root of 3 over 2")[$display(frac(sqrt(3), 2))$]


        
      
      
        #math.equation(block: false, alt: "f open parenthesis the fraction θ over 2 close parenthesis")[$f ( display(frac(θ, 2)) )$] for #math.equation(block: false, alt: "θ equals 120 degrees")[$θ = 120^(∘)$]


      
      
        #math.equation(block: false, alt: "g open parenthesis 4 θ close parenthesis minus f open parenthesis 2 θ close parenthesis")[$g ( 4 θ ) − f ( 2 θ )$] for #math.equation(block: false, alt: "θ equals 15 degrees")[$θ = 15^(∘)$]


        
          0


        
      
      
        #math.equation(block: false, alt: "the fraction 1 minus g open parenthesis 2 θ close parenthesis over 2")[$display(frac(1 − g ( 2 θ ), 2))$] for #math.equation(block: false, alt: "θ equals 45 degrees")[$θ = 45^(∘)$]


      
      For Problems 43–46, write an equation for the given graph, and give the exact coordinates of the labeled points.


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


        
          #math.equation(block: true, alt: "y equals 1.5 cos ⁡ open parenthesis the fraction θ over 3 close parenthesis , M open parenthesis minus 90 degrees , the fraction 3 the square root of 3 over 4 close parenthesis , N open parenthesis 180 degrees , the fraction 3 over 4 close parenthesis")[$y = 1.5 cos ( display(frac(θ, 3)) ) , " " M ( − 90^(∘) , display(frac(3 sqrt(3), 4)) ) , N ( 180^(∘) , display(frac(3, 4)) )$]


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


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


        
          #math.equation(block: true, alt: "y equals 3 plus 3 sin ⁡ 2 θ , A open parenthesis minus 45 degrees , 6 close parenthesis , B open parenthesis 120 degrees , 3 minus the fraction 3 the square root of 3 over 2 close parenthesis")[$y = 3 + 3 sin 2 θ , " " A ( − 45^(∘) , 6 ) , B ( 120^(∘) , 3 − display(frac(3 sqrt(3), 2)) )$]


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


      
      
        Every 24 hours Delbert takes 50 mg of a therapeutic drug.  The level of that drug in Delbert's bloodstream immediately jumps to its peak level of 60 mg, but the level diminishes to its lowest level of 10 mg just before the next dose.


        + Sketch a graph of #math.equation(block: false, alt: "d open parenthesis t close parenthesis")[$d ( t )$], the amount of the drug in Delbert's bloodstream  after one of his doses, Show at least three doses.
+ What is the period of #math.equation(block: false, alt: "d open parenthesis t close parenthesis")[$d ( t )$]?


        
          + #figure(figph[periodic graph], alt: "periodic graph", caption: none)
+ 24 hours


        
      
      
        A water fountain has water trickling into a container, but once the container is full, it tilts and the water pours quickly out.  Then the container tilts back and starts to fill again.  The container is filled 5 times every minute.


        + Sketch a graph of #math.equation(block: false, alt: "h open parenthesis t close parenthesis")[$h ( t )$], the height of water in the container at time Show at least two complete fill-and-empty cycles.
+ What is the period of #math.equation(block: false, alt: "h open parenthesis t close parenthesis")[$h ( t )$]?


      
      
        Henry is watching Billie ride a carousel.  He stands 2 meters from the carousel, which has a diameter of 10 meters.  He notices that she passes by him three times each minute.


        + Sketch a graph of #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$], the distance between Henry and Billie at time #math.equation(block: false, alt: "t")[$t$]. Show at least two complete circuits.
+ What is the period of #math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$]?


        
          + #figure(figph[periodic graph], alt: "periodic graph", caption: none)
+ 20 sec


        
      
      
        An ant walks at constant unit speed (1 unit of distance per second) along the triangle with vertices at #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis , open parenthesis 1 , 1 close parenthesis ,")[$( 0 , 0 ) , " " ( 1 , 1 ) ,$] and #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$].


        + How far does the ant need to walk to get from #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$] to #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$]?  From #math.equation(block: false, alt: "open parenthesis 1 , 1 close parenthesis")[$( 1 , 1 )$] to #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$]?  From #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] to #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis")[$( 0 , 0 )$]?
+ Sketch a graph of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$], the ant's #math.equation(block: false, alt: "y")[$y$]-coordinate at time #math.equation(block: false, alt: "t")[$t$]. Show at least two circuits around the triangle.
+ What is the period of #math.equation(block: false, alt: "g open parenthesis t close parenthesis")[$g ( t )$]?


      
      For Problems 51–54,


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


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


        
          + #figure(figph[y = 4 + 2 cos theta], alt: "y = 4 + 2 cos theta", caption: none)
+ amp: 2, period: #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], midline: #math.equation(block: false, alt: "y equals 4")[$y = 4$]


        
      
      
        #math.equation(block: true, alt: "y equals minus 1 plus 3 cos ⁡ θ")[$y = − 1 + 3 cos θ$]


      
      
        #math.equation(block: true, alt: "y equals 1.5 plus 3.5 sin ⁡ 2 θ")[$y = 1.5 + 3.5 sin 2 θ$]


        
          + #figure(figph[y = 1.5 + 3.5 sin (2 theta)], alt: "y = 1.5 + 3.5 sin (2 theta)", caption: none)
+ amp: 3.5, period: #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], midline: #math.equation(block: false, alt: "y equals 1.5")[$y = 1.5$]


        
      
      
        #math.equation(block: true, alt: "y equals 1.6 plus 1.4 sin ⁡ open parenthesis 0.5 θ close parenthesis")[$y = 1.6 + 1.4 sin ( 0.5 θ )$]


      
      For Problems 55–58, find the angle of inclination of the line.


      
        #math.equation(block: true, alt: "y equals the fraction the square root of 3 over 3 x plus 1")[$y = display(frac(sqrt(3), 3)) x + 1$]


        
          #math.equation(block: true, alt: "30 degrees")[$30^(∘)$]


        
      
      
        #math.equation(block: true, alt: "y equals minus x minus 11")[$y = − x − 11$]


      
      
        #math.equation(block: true, alt: "y equals 100 minus 28 x")[$y = 100 − 28 x$]


        
          #math.equation(block: true, alt: "92.05 degrees")[$92.05^(∘)$]


        
      
      
        #math.equation(block: true, alt: "y equals minus 3.7 plus 1.4 x")[$y = − 3.7 + 1.4 x$]


      
      For Problems 59–62, find an equation for the line passing through the given point with  angle of inclination #math.equation(block: false, alt: "α")[$α$].


      
        #math.equation(block: true, alt: "open parenthesis 0 , 2 close parenthesis")[$( 0 , 2 )$], #math.equation(block: true, alt: "α equals 45 degrees")[$" " α = 45^(∘)$]


        
          #math.equation(block: true, alt: "y equals x plus 2")[$y = x + 2$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$], #math.equation(block: true, alt: "α equals 135 degrees")[$" " α = 135^(∘)$]


      
      
        #math.equation(block: true, alt: "open parenthesis 3 , minus 4 close parenthesis")[$( 3 , − 4 )$], #math.equation(block: true, alt: "α equals 120 degrees")[$" " α = 120^(∘)$]


        
          #math.equation(block: true, alt: "y equals minus the square root of 3 x plus 3 the square root of 3 minus 4")[$y = − sqrt(3) x + 3 sqrt(3) − 4$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis minus 7 , 2 close parenthesis")[$( − 7 , 2 )$], #math.equation(block: true, alt: "α equals 60 degrees")[$" " α = 60^(∘)$]


      
      
        Sketch the graphs of #math.equation(block: false, alt: "y equals tan ⁡ θ")[$y = tan θ$] and #math.equation(block: false, alt: "y equals cos ⁡ θ")[$y = cos θ$] on the same grid for #math.equation(block: false, alt: "minus 180 degrees less than θ less than 180 degrees")[$− 180^(∘) < θ < 180^(∘)$].  How are the #math.equation(block: false, alt: "θ")[$θ$]-intercepts of the graph of #math.equation(block: false, alt: "y equals cos ⁡ θ")[$y = cos θ$] related to the graph of #math.equation(block: false, alt: "y equals tan ⁡ θ")[$y = tan θ$] ?


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


          The #math.equation(block: false, alt: "θ")[$θ$]-intercepts of #math.equation(block: false, alt: "cos ⁡ θ")[$cos θ$] occur at the vertical asymptotes of #math.equation(block: false, alt: "tan ⁡ θ")[$tan θ$].


        
      
      
        Sketch the graphs of #math.equation(block: false, alt: "y equals tan ⁡ θ")[$y = tan θ$] and #math.equation(block: false, alt: "y equals sin ⁡ θ")[$y = sin θ$] on the same grid for #math.equation(block: false, alt: "minus 180 degrees less than θ less than 180 degrees")[$− 180^(∘) < θ < 180^(∘)$].  How are the #math.equation(block: false, alt: "θ")[$θ$]-intercepts of the graph of #math.equation(block: false, alt: "y equals sin ⁡ θ")[$y = sin θ$] related to the graph of #math.equation(block: false, alt: "y equals tan ⁡ θ")[$y = tan θ$] ?