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== 4.1#h(0.6em)Angles and Rotation


    
      
=== Introduction


      So far we have studied angles as parts of triangles, but we can also use angles to describe rotation. For example, think of the minute hand on a clock.  Every hour, the minute hand moves through one complete rotation, or #math.equation(block: false, alt: "360 degrees")[$360^(∘)$]. In two hours, the minute hand rotates through #math.equation(block: false, alt: "720 degrees")[$720^(∘)$].


      #examplebox("Example 1")[][
Through how many degrees does the minute hand rotate in an hour and a half? In forty minutes?


        
        #solutionbox[
Look at the figure at right. An hour and a half represents 1.5 complete rotations, or


          #math.equation(block: true, alt: "1.5 open parenthesis 360 degrees close parenthesis equals 540 degrees")[$1.5 ( 360^(∘) ) = 540^(∘)$]
          #figure(figph[clock], alt: "clock", caption: none)


          Forty minutes is two-thirds of an hour, so the minute hand rotates through


          #math.equation(block: true, alt: "the fraction 2 over 3 open parenthesis 360 degrees close parenthesis equals 240 degrees")[$display(frac(2, 3)) ( 360^(∘) ) = 240^(∘)$]
          #figure(figph[clock], alt: "clock", caption: none)
]
]


      
        The volume control on an amplifier is a dial with ten settings, as shown at right. Through how many degrees would you rotate the dial to increase the volume level from 0 to 7?


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


        
          #math.equation(block: false, alt: "0.7")[$0.7$] of #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], or #math.equation(block: false, alt: "252 degrees")[$252^(∘)$]


        
      
    
    
      
=== Angles in Standard Position


      The degree measure of an angle depends only on the fraction of a whole rotation between its sides, and not on the location or position of the angle.  To compare and analyze angles, we place them in #strong[standard position], so that the vertex of the angle is located at the origin and its initial side lies on the positive #math.equation(block: false, alt: "x")[$x$]-axis. If the terminal side rotates counter-clockwise, the degree measure of the angle is positive; it is negative if the terminal side rotates clockwise. The figure below shows four angles and how each appears in standard position.


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


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


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


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


      One-half a complete revolution is #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], and three-quarters of one revolution is #math.equation(block: false, alt: "270 degrees")[$270^(∘)$]. Thus, for angles between #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] and #math.equation(block: false, alt: "270 degrees")[$270^(∘)$] in standard position, the terminal side lies in the third quadrant, and for angles between #math.equation(block: false, alt: "270 degrees")[$270^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], the terminal side lies in the fourth quadrant.


      #examplebox("Example 2")[][
Find the degree measure of the angles #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "α")[$α$] shown below, and sketch each angle in standard position.


          #figure(figph[Regular pentagon inscribed in circle], alt: "Regular pentagon inscribed in circle", caption: none)


          #figure(figph[Clock with minute hand pointing to 12 and to 11], alt: "Clock with minute hand pointing to 12 and to 11", caption: none)


        
        #solutionbox[
+ The angle #math.equation(block: false, alt: "α")[$α$] is one-fifth of a complete revolution, or 
      #math.equation(block: true, alt: "the fraction 1 over 5 open parenthesis 360 degrees close parenthesis equals 72 degrees")[$display(frac(1, 5)) ( 360^(∘) ) = 72^(∘)$]  
    In standard position, it is a first-quadrant angle, as shown in figure (a) below.#figure(figph[72 degree angle in standard position], alt: "72 degree angle in standard position", caption: none)

#figure(figph[330 degree angle in standard position], alt: "330 degree angle in standard position", caption: none)
+ The angle #math.equation(block: false, alt: "β")[$β$] is #math.equation(block: false, alt: "the fraction 11 over 12")[$display(frac(11, 12))$] of a complete revolution, or 
      #math.equation(block: true, alt: "the fraction 11 over 12 open parenthesis 360 degrees close parenthesis equals 330 degrees")[$display(frac(11, 12)) ( 360^(∘) ) = 330^(∘)$] 
  In standard position, it is a fourth-quadrant angle. (See Figure (b).)
]
]


      
        Find the degree measure of each angle below, and sketch the angle in standard position.


        #figure(figph[Circle divided into 3 equal sectors], alt: "Circle divided into 3 equal sectors", caption: none)


        #figure(figph[angle difference between 250 degrees and straight angle], alt: "angle difference between 250 degrees and straight angle", caption: none)


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


          #figure(figph[120 degree angle in standard position], alt: "120 degree angle in standard position", caption: none)


          b. #math.equation(block: false, alt: "70 degrees")[$70^(∘)$]


          #figure(figph[70 degree angle in standard position], alt: "70 degree angle in standard position", caption: none)


        
      
    
    
      
=== Trigonometric Ratios for All Angles


      In Chapter 3 we defined the sine, cosine, and tangent for obtuse angles by placing the angle in a Cartesian coordinate system. We can do the same for angles that represent rotations.


      - First, we place the angle #math.equation(block: false, alt: "θ")[$θ$] in standard position, with its vertex at the origin. We picture the terminal side sweeping counter-clockwise around a circle to form the angle.
- Next, we choose a point #math.equation(block: false, alt: "P")[$P$] with coordinates #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] on the terminal side, as shown at right.  The distance from the origin to #math.equation(block: false, alt: "P")[$P$] is then #math.equation(block: false, alt: "r equals the square root of x squared plus y squared")[$r = sqrt(x^(2) + y^(2))$]. The trigonometric ratios of #math.equation(block: false, alt: "θ")[$θ$] are defined as follows.


      #figure(figph[Reference triangle in third quadrant], alt: "Reference triangle in third quadrant", caption: none)


      #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 ⁡ open parenthesis θ close parenthesis equals the fraction y over r cos ⁡ open parenthesis θ close parenthesis equals the fraction x over r tan ⁡ open parenthesis θ close parenthesis equals the fraction y over x")[$bold(sin) ( bold(italic(θ)) ) = display(frac(bold(italic(y)), bold(italic(r)))) " " " " " " " " " " " " " " " " " " bold(cos) ( bold(italic(θ)) ) = display(frac(bold(italic(x)), bold(italic(r)))) " " " " " " " " " " " " " " " " " " bold(tan) ( bold(italic(θ)) ) = display(frac(bold(italic(y)), bold(italic(x))))$]
]


      We can choose any point on the terminal side of the angle, and the trig ratios defined by its coordinates will be the same. (Can you explain why? Think about similar triangles.) And, as we noted when we first defined the trig ratios, they are functions of the input angle, so there is only one sine, cosine, or tangent for a given angle.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The notation #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$] is read "sine of theta," just as we say "f of x" for #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]. The parentheses do not mean multiplication, but rather tell us to apply the function.
]


      Because it is the distance from the origin to #math.equation(block: false, alt: "P")[$P$], #math.equation(block: false, alt: "r")[$r$] is always positive. However, #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] can be positive or negative (or zero), depending on the angle #math.equation(block: false, alt: "θ")[$θ$]. For example, in the second quadrant, #math.equation(block: false, alt: "x")[$x$] is negative but #math.equation(block: false, alt: "y")[$y$] is positive, so the cosine and the tangent of angles between #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] and #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] are negative, but their sines are positive.


      #examplebox("Example 3")[][
Give the sign of each of the three trigonometric ratios of the angles.


          + #math.equation(block: false, alt: "200 degrees")[$200^(∘)$]
+ #math.equation(block: false, alt: "300 degrees")[$300^(∘)$]


        
        #solutionbox[
a.  In standard position, the terminal side of an angle of #math.equation(block: false, alt: "200 degrees")[$200^(∘)$] lies in the third quadrant. (See figure (a) below.) In the third quadrant, #math.equation(block: false, alt: "x less than 0")[$x < 0$] and #math.equation(block: false, alt: "y less than 0")[$y < 0$], but #math.equation(block: false, alt: "r greater than 0")[$r > 0$].  Thus, #math.equation(block: false, alt: "sin ⁡ open parenthesis 200 degrees close parenthesis")[$sin ( 200^(∘) )$] is negative, #math.equation(block: false, alt: "cos ⁡ open parenthesis 200 degrees close parenthesis")[$cos ( 200^(∘) )$] is negative, and #math.equation(block: false, alt: "tan ⁡ open parenthesis 200 degrees close parenthesis")[$tan ( 200^(∘) )$] is positive.


          #figure(figph[200 deg angle], alt: "200 deg angle", caption: none)


          #figure(figph[300 deg angle], alt: "300 deg angle", caption: none)


          b.  The terminal side of #math.equation(block: false, alt: "300 degrees")[$300^(∘)$] lies in the fourth quadrant, so #math.equation(block: false, alt: "x greater than 0")[$x > 0$] and #math.equation(block: false, alt: "y less than 0")[$y < 0$], and #math.equation(block: false, alt: "r greater than 0")[$r > 0$].  Thus, #math.equation(block: false, alt: "sin ⁡ open parenthesis 300 degrees close parenthesis")[$sin ( 300^(∘) )$] is negative, #math.equation(block: false, alt: "cos ⁡ open parenthesis 300 degrees close parenthesis")[$cos ( 300^(∘) )$] is positive, and #math.equation(block: false, alt: "tan ⁡ open parenthesis 300 degrees close parenthesis")[$tan ( 300^(∘) )$] is negative.
]
]


      
        For angles in each of the four quadrants shown below, explain why the indicated trig ratios are positive. Then complete the table.


        #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Quadrant], [Degrees], [Sine], [Cosine], [Tangent]),
  [First], [#math.equation(block: false, alt: "0 degrees less than θ less than 90 degrees")[$0^(∘) < θ < 90^(∘)$]], [positive], [positive], [positive],
  [Second], [#math.equation(block: false, alt: "90 degrees less than θ less than 180 degrees")[$90^(∘) < θ < 180^(∘)$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [Third], [#math.equation(block: false, alt: "180 degrees less than θ less than 270 degrees")[$180^(∘) < θ < 270^(∘)$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [Fourth], [#math.equation(block: false, alt: "270 degrees less than θ less than 360 degrees")[$270^(∘) < θ < 360^(∘)$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


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


        
          #figure(table(
  columns: 5, align: left, inset: 6pt,
  table.header([Quadrant], [Degrees], [Sine], [Cosine], [Tangent]),
  [First], [#math.equation(block: false, alt: "0 degrees less than θ less than 90 degrees")[$0^(∘) < θ < 90^(∘)$]], [positive], [positive], [positive],
  [Second], [#math.equation(block: false, alt: "90 degrees less than θ less than 180 degrees")[$90^(∘) < θ < 180^(∘)$]], [positive], [negative], [negative],
  [Third], [#math.equation(block: false, alt: "180 degrees less than θ less than 270 degrees")[$180^(∘) < θ < 270^(∘)$]], [negative], [negative], [positive],
  [Fourth], [#math.equation(block: false, alt: "270 degrees less than θ less than 360 degrees")[$270^(∘) < θ < 360^(∘)$]], [negative], [positive], [negative],
))


        
      
      #examplebox("Example 4")[][
Find the sine, cosine, and tangent of the angle shown at right.


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


        
        #solutionbox[
The #math.equation(block: false, alt: "y")[$y$]-coordinate of the point #math.equation(block: false, alt: "P")[$P$] is #math.equation(block: false, alt: "minus 5")[$− 5$], and #math.equation(block: false, alt: "r equals 6")[$r = 6$], so


          #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis equals the fraction y over r equals the fraction minus 5 over 6")[$sin ( θ ) = display(frac(y, r)) = display(frac(− 5, 6))$]
          To find the #math.equation(block: false, alt: "x")[$x$]-coordinate of #math.equation(block: false, alt: "P")[$P$], we use the equation of a circle of radius #math.equation(block: false, alt: "6")[$6$], #math.equation(block: false, alt: "x squared plus y squared equals 36")[$x^(2) + y^(2) = 36$].


          #math.equation(block: true, alt: "x squared plus open parenthesis minus 5 close parenthesis squared, equals 36; x squared, equals 36 minus 25 equals 11; x, equals plus or minus the square root of 11")[$x^(2) + ( − 5 )^(2) & = 36 \ x^(2) & = 36 − 25 = 11 \ x & = ± sqrt(11)$]
          Because #math.equation(block: false, alt: "P")[$P$] is in the third quadrant, #math.equation(block: false, alt: "x equals minus the square root of 11")[$x = − sqrt(11)$]. Thus,


          #math.equation(block: true, alt: "cos ⁡ open parenthesis θ close parenthesis equals the fraction x over r equals the fraction minus the square root of 11 over 6 and tan ⁡ open parenthesis θ close parenthesis equals the fraction y over x equals the fraction minus 5 over minus the square root of 11 equals the fraction 5 over the square root of 11")[$cos ( θ ) = display(frac(x, r)) = display(frac(− sqrt(11), 6)) " " " " " " " " "and" " " " " " " " " tan ( θ ) = display(frac(y, x)) = display(frac(− 5, − sqrt(11))) = display(frac(5, sqrt(11)))$]
]
]


      
        Find the sine, cosine, and tangent of the angle shown at right. The circle has radius 4.


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


        
          We know that #math.equation(block: false, alt: "r equals 4")[$r = 4$] and #math.equation(block: false, alt: "x equals minus 3")[$x = − 3$], so


          #math.equation(block: true, alt: "y equals the square root of 4 squared minus open parenthesis minus 3 close parenthesis squared equals the square root of 7")[$y = sqrt(4^(2) − ( − 3 )^(2)) = sqrt(7)$]
          Thus,


          #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis equals the fraction the square root of 7 over 4 , cos ⁡ open parenthesis θ close parenthesis equals the fraction minus 3 over 4 , tan ⁡ open parenthesis θ close parenthesis equals the fraction minus the square root of 7 over 3")[$sin ( θ ) = display(frac(sqrt(7), 4)) , " " " " cos ( θ ) = display(frac(− 3, 4)) , " " " " tan ( θ ) = display(frac(− sqrt(7), 3))$]
        
      
    
    
      
=== Reference Angles


      In Section 3.1 we learned that the trig ratios for angles in the second quadrant are the same as the trig ratios for their supplements, except for sign. For example, you can use your calculator to verify that


      #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "sin ⁡ open parenthesis 130 degrees close parenthesis equals 0.7660")[$sin ( 130^(∘) ) = 0.7660$]], [#math.equation(block: false, alt: "sin ⁡ open parenthesis 50 degrees close parenthesis equals 0.7660")[$sin ( 50^(∘) ) = 0.7660$]]),
  [#math.equation(block: false, alt: "cos ⁡ open parenthesis 130 degrees close parenthesis equals minus 0.6428")[$cos ( 130^(∘) ) = − 0.6428$]], [#math.equation(block: false, alt: "cos ⁡ open parenthesis 50 degrees close parenthesis equals 0.6428")[$cos ( 50^(∘) ) = 0.6428$]],
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis 130 degrees close parenthesis equals minus 1.1918")[$tan ( 130^(∘) ) = − 1.1918$]], [#math.equation(block: false, alt: "tan ⁡ open parenthesis 50 degrees close parenthesis equals 1.1918")[$tan ( 50^(∘) ) = 1.1918$]],
))


      #figure(figph[semi-circle with angles], alt: "semi-circle with angles", caption: none)


      The trig ratios for #math.equation(block: false, alt: "130 degrees")[$130^(∘)$] and #math.equation(block: false, alt: "50 degrees")[$50^(∘)$] have the same absolute value because the two triangles formed by the angles are congruent, as shown above.


      #math.equation(block: false, alt: "△ O P prime Q prime")[$upright(△) O P^(′) Q^(′)$] is called a #strong[reference triangle] for #math.equation(block: false, alt: "130 degrees")[$130^(∘)$], and #math.equation(block: false, alt: "50 degrees")[$50^(∘)$] is called the #strong[reference angle].


      The trig ratios for angles between #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], whose terminal sides lie in the third and fourth quadrants, are also related to the trig ratios of familiar angles in the first quadrant. We "refer" the angle to a first quadrant angle with a congruent reference triangle.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Reference angles  will be important when we need to solve trigonometric equations.  There will always be two angles with the same trig ratio (except for quadrantal angles), and they will have the same reference angle.
]


      We can construct reference triangles for angles in any of the four quadrants, and the trig ratios of the angle are the same as the trig ratios of its reference angle, up to sign. Here is how to construct a reference triangle for an angle :


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Constructing a reference triangle]


        + 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 figure below shows angles #math.equation(block: false, alt: "θ")[$θ$] between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], and the reference angle, #math.equation(block: false, alt: "θ tilde")[$tilde(θ)$], for each. Study the figures, and make sure you understand the formula for finding the reference angle in each quadrant.


      #figure(figph[Reference angles for all quadrants], alt: "Reference angles for all quadrants", caption: none)


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
These observations may help you remember the formulas:


        - The right triangle formed in this way always lies between the terminal side 
        and the #math.equation(block: false, alt: "x")[$x$]-axis.
- The positive acute angle formed between the terminal side and the #math.equation(block: false, alt: "x")[$x$]-axis 
        is the reference angle, and the right triangle is the reference triangle.
]


      #notebox("Activity", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Reference Angles]


        The circle below has radius 1 and is centered at the origin.


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


        + + Use a protractor to draw an angle of #math.equation(block: false, alt: "56 degrees")[$56^(∘)$] in standard position. Draw its reference triangle.
+ Use your calculator to find the sine and cosine of #math.equation(block: false, alt: "56 degrees")[$56^(∘)$], rounded to two decimal places.  
    Label the sides of the reference triangle with their lengths.
+ What are the coordinates of the point #math.equation(block: false, alt: "P")[$P$] where your angle intersects the circle?
+ Draw the reflection of your reference triangle across the #math.equation(block: false, alt: "y")[$y$]-axis, so that you have a 
    congruent triangle in the second quadrant.
+ + You now have the reference triangle for a second-quadrant angle in standard position.  
    What is that angle?
+ Use your calculator to find the sine and cosine of your new angle. Label the coordinates 
    of the point #math.equation(block: false, alt: "Q")[$Q$] where the angle intersects the circle.
+ Draw the reflection of your triangle from part (1d) across the #math.equation(block: false, alt: "x")[$x$]-axis, so that you have a 
    congruent triangle in the third quadrant.
+ + You now have the reference triangle for a third-quadrant angle in standard position.  
    What is that angle?
+ Use your calculator to find the sine and cosine of your new angle. Label the coordinates 
    of the point #math.equation(block: false, alt: "R")[$R$] where the angle intersects the circle.
+ Draw the reflection of your triangle from part (2c) across the #math.equation(block: false, alt: "y")[$y$]-axis, so that you have a 
    congruent triangle in the fourth quadrant.
+ + You now have the reference triangle for a fourth-quadrant angle in standard position.  
    What is that angle?
+ Use your calculator to find the sine and cosine of your new angle.  Label the coordinates 
    of the point where the angle intersects the circle.


        Generalize:  All four of your angles have the same reference angle, #math.equation(block: false, alt: "56 degrees")[$56^(∘)$]. For each quadrant, write a formula for the angle whose reference angle is #math.equation(block: false, alt: "θ")[$θ$].


        - Quadrant I:
- Quadrant II:
- Quadrant III:
- Quadrant IV:
]


      #examplebox("Example 5")[][
+ Find the reference angle for #math.equation(block: false, alt: "200 degrees")[$200^(∘)$].
+ Sketch #math.equation(block: false, alt: "200 degrees")[$200^(∘)$] and its reference angle in standard position, along with their reference triangles. Verify that both angles have the same trigonometric ratios, up to sign.


        
        #solutionbox[
+ In standard position, an angle of #math.equation(block: false, alt: "200 degrees")[$200^(∘)$] lies in the third quadrant. Its reference angle is 
        #math.equation(block: true, alt: "200 degrees minus 180 degrees equals 20 degrees")[$200^(∘) − 180^(∘) = 20^(∘)$]
+ Both angles are shown at right. Note that the reference triangle for #math.equation(block: false, alt: "200 degrees")[$200^(∘)$] is congruent to the reference triangle for #math.equation(block: false, alt: "20 degrees")[$20^(∘)$]. You can use your calculator to verify the following values.


          #figure(figph[200 degree and reference angles], alt: "200 degree and reference angles", caption: none)


          #math.equation(block: true, alt: "sin ⁡ open parenthesis 20 degrees close parenthesis, equals 0.3420, sin ⁡ open parenthesis 200 degrees close parenthesis equals minus 0.3420; cos ⁡ open parenthesis 20 degrees close parenthesis, equals 0.9397, cos ⁡ open parenthesis 200 degrees close parenthesis equals minus 0.9397; tan ⁡ open parenthesis 20 degrees close parenthesis, equals 0.3640, tan ⁡ open parenthesis 200 degrees close parenthesis equals 0.3640")[$sin ( 20^(∘) ) & = 0.3420 &  & sin ( 200^(∘) ) = − 0.3420 \ cos ( 20^(∘) ) & = 0.9397 &  & cos ( 200^(∘) ) = − 0.9397 \ tan ( 20^(∘) ) & = 0.3640 &  & tan ( 200^(∘) ) = 0.3640$]
]
]


      
        + Find the reference angle for #math.equation(block: false, alt: "285 degrees")[$285^(∘)$].
+ Sketch #math.equation(block: false, alt: "285 degrees")[$285^(∘)$] and its reference angle in standard position, along with their reference triangles. Verify that both angles have the same trigonometric ratios, up to sign.


        
          + The terminal side of an angle of #math.equation(block: false, alt: "285 degrees")[$285^(∘)$] lies in the fourth quadrant, so its reference angle is #math.equation(block: false, alt: "360 minus 285 equals 75 degrees")[$360 − 285 = 75^(∘)$]
+ #figure(figph[285 degree and reference angles], alt: "285 degree and reference angles", caption: none)

#math.equation(block: true, alt: "sin ⁡ open parenthesis 285 degrees close parenthesis equals minus sin ⁡ open parenthesis 75 degrees close parenthesis equals minus 0.9659")[$sin ( 285^(∘) ) = − sin ( 75^(∘) ) = − 0.9659$] #math.equation(block: true, alt: "cos ⁡ open parenthesis 285 degrees close parenthesis equals cos ⁡ open parenthesis 75 degrees close parenthesis equals 0.2588")[$cos ( 285^(∘) ) = cos ( 75^(∘) ) = 0.2588$] #math.equation(block: true, alt: "tan ⁡ open parenthesis 285 degrees close parenthesis equals minus tan ⁡ open parenthesis 75 degrees close parenthesis equals minus 3.7321")[$tan ( 285^(∘) ) = − tan ( 75^(∘) ) = − 3.7321$]


        
      
    
    
      
=== Using Reference Angles


      Here is a summary of our discussion about reference angles.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Reference Angles]


        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.
]


      Any acute angle #math.equation(block: false, alt: "θ")[$θ$] is the reference angle for four angles between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], one in each quadrant. The figure below shows the four angles in standard position whose reference angle is #math.equation(block: false, alt: "35 degrees")[$35^(∘)$]. Note that each angle is found by measuring #math.equation(block: false, alt: "35 degrees")[$35^(∘)$] from the #math.equation(block: false, alt: "x")[$x$]-axis in the appropriate quadrant, and that the four angles together make a “bow-tie” shape.


      #figure(figph[angleswith ref angle 35 degrees], alt: "angleswith ref angle 35 degrees", caption: none)


      From the figure, you can see that the angles in each quadrant with a given reference angle are computed as follows.


      #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)$]],
))
]


      #examplebox("Example 6")[][
+ Find two angles between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] whose cosine is #math.equation(block: false, alt: "the fraction 5 over 8")[$display(frac(5, 8))$].  Round your answers to the nearest tenth of a degree.
+ Find two angles between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] whose cosine is #math.equation(block: false, alt: "the fraction minus 5 over 8")[$display(frac(− 5, 8))$].  Sketch the reference triangle for each angle.


        
        #solutionbox[
+ One of the angles we want is #math.equation(block: false, alt: "cos to the power minus 1 ⁡ open parenthesis the fraction 5 over 8 close parenthesis")[$cos^(− 1) ( display(frac(5, 8)) )$]. Use your calculator to find the first quadrant angle, #math.equation(block: false, alt: "51.3 degrees")[$51.3^(∘)$]. The cosine is also positive in the fourth quadrant, so we look for the angle in the fourth quadrant with reference angle #math.equation(block: false, alt: "51.3 degrees")[$51.3^(∘)$]. That angle is #math.equation(block: false, alt: "360 degrees minus 51.3 degrees equals 308.7 degrees")[$360^(∘) − 51.3^(∘) = 308.7^(∘)$], as shown at right.

#figure(figph[angles], alt: "angles", caption: none)
+ Because the cosine is negative in the second and third quadrants, we would like angles #math.equation(block: false, alt: "θ")[$θ$] in those quadrants whose reference angle is #math.equation(block: false, alt: "θ tilde equals 51.3 degrees")[$tilde(θ) = 51.3^(∘)$]. In the second quadrant, we find

#math.equation(block: true, alt: "θ, equals 180 degrees minus θ tilde; equals 180 degrees minus 51.3 degrees equals 128.7 degrees")[$θ & = 180^(∘) − tilde(θ) \  & = 180^(∘) − 51.3^(∘) = 128.7^(∘)$]And in the third quadrant,

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

 #linebreak() #math.equation(block: true, alt: "θ, equals 180 degrees plus θ tilde; equals 180 degrees plus 51.3 degrees equals 231.3 degrees")[$θ & = 180^(∘) + tilde(θ) \  & = 180^(∘) + 51.3^(∘) = 231.3^(∘)$]
    Both angles are shown above. You can check that the cosines of #math.equation(block: false, alt: "128.7 degrees")[$128.7^(∘)$] and #math.equation(block: false, alt: "231.3 degrees")[$231.3^(∘)$] are both approximately #math.equation(block: false, alt: "minus 0.625")[$− 0.625$].
]
]


      
        + Find an angle in the third quadrant whose tangent is #math.equation(block: false, alt: "3.66")[$3.66$]. Round your answer to the nearest tenth of a degree.
+ Use reference angles to find two angles whose tangent is #math.equation(block: false, alt: "minus 3.66")[$− 3.66$].


        
          + #math.equation(block: false, alt: "tan to the power minus 1 ⁡ open parenthesis 3.66 close parenthesis equals 74.7 degrees")[$tan^(− 1) ( 3.66 ) = 74.7^(∘)$]. The angle in the third quadrant with reference angle #math.equation(block: false, alt: "74.7 degrees")[$74.7^(∘)$] is #math.equation(block: false, alt: "180 degrees plus 74.7 degrees equals 254.7 degrees")[$180^(∘) + 74.7^(∘) = 254.7^(∘)$]
+ The tangent is negative in the second and fourth quadrants, and the angles in those quadrants with reference angle #math.equation(block: false, alt: "74.7 degrees")[$74.7^(∘)$] are #math.equation(block: false, alt: "105.3 degrees")[$105.3^(∘)$] and #math.equation(block: false, alt: "285.3 degrees")[$285.3^(∘)$]


        
      
    
    
      
=== The Special Angles


      Recall that the angles #math.equation(block: false, alt: "30 degrees , 45 degrees")[$30^(∘) , " " 45^(∘)$] and #math.equation(block: false, alt: "60 degrees")[$60^(∘)$] are called the special angles because we can express the exact values of their trigonometric ratios in terms of radicals. There are special angles in all four quadrants; namely, those whose reference angles are #math.equation(block: false, alt: "30 degrees , 45 degrees")[$30^(∘) , " " 45^(∘)$] and #math.equation(block: false, alt: "60 degrees")[$60^(∘)$].


      #examplebox("Example 7")[][
Find exact values for the sine, cosine, and tangent of #math.equation(block: false, alt: "210 degrees")[$210^(∘)$].


        
        #solutionbox[
An angle of #math.equation(block: false, alt: "210 degrees")[$210^(∘)$] lies in the third quadrant, and its reference angle is


          #math.equation(block: true, alt: "θ tilde equals 210 degrees minus 180 degrees equals 30 degrees")[$tilde(θ) = 210^(∘) − 180^(∘) = 30^(∘)$]
          In the third quadrant, the sine and cosine are negative, so we have


          #math.equation(block: true, alt: "sin ⁡ open parenthesis θ close parenthesis equals the fraction minus 1 over 2 , cos ⁡ open parenthesis θ close parenthesis equals the fraction minus the square root of 3 over 2 , tan ⁡ open parenthesis θ close parenthesis equals the fraction 1 over the square root of 3")[$sin ( θ ) = display(frac(− 1, 2)) , " " " " " " cos ( θ ) = display(frac(− sqrt(3), 2)) , " " " " " " tan ( θ ) = display(frac(1, sqrt(3)))$]
          Alternatively, consider the reference triangle in the figure at right, where #math.equation(block: false, alt: "r equals 2")[$r = 2$]. Because the sides of a 30-60-90 triangle are in the ratio #math.equation(block: false, alt: "1 : the square root of 3 : 2")[$1 : sqrt(3) : 2$], the point #math.equation(block: false, alt: "P")[$P$] has coordinates #math.equation(block: false, alt: "open parenthesis minus the square root of 3 , minus 1 close parenthesis")[$( − sqrt(3) , − 1 )$]. Using the definitions of the trig ratios, we can calculate the values above.


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


      
        + Sketch an angle of #math.equation(block: false, alt: "300 degrees")[$300^(∘)$] in standard position, and its reference triangle. Find the reference angle for #math.equation(block: false, alt: "300 degrees")[$300^(∘)$].
+ Find exact values for the sine, cosine, and tangent of #math.equation(block: false, alt: "300 degrees")[$300^(∘)$].


        
          + #figure(figph[Reference angle for 300 degrees], alt: "Reference angle for 300 degrees", caption: none)

Reference angle:  #math.equation(block: false, alt: "60 degrees")[$" " 60^(∘)$]
+ #math.equation(block: false, alt: "sin ⁡ open parenthesis 300 degrees close parenthesis equals the fraction minus the square root of 3 over 2")[$sin ( 300^(∘) ) = display(frac(− sqrt(3), 2))$], #math.equation(block: false, alt: "cos ⁡ open parenthesis 300 degrees close parenthesis equals the fraction 1 over 2")[$" " cos ( 300^(∘) ) = display(frac(1, 2))$], #math.equation(block: false, alt: "tan ⁡ open parenthesis 300 degrees close parenthesis equals minus the square root of 3")[$" " tan ( 300^(∘) ) = − sqrt(3)$]


        
      
      All of the special angles are shown at right. In the Homework Problems you will calculate the three trigonometric ratios for all the special angles, and it will be useful to be familiar with these values, and be able to calculate them quickly. You may want to review the two "special triangles" in Section 2.3.


      #figure(figph[Special angles on unit circle], alt: "Special angles on unit circle", caption: none)


      
        
==== Coterminal Angles


        Because #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] represents one complete revolution, we can add or subtract a multiple of #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] to any angle, and the terminal side will arrive at the same position. For example, the angles #math.equation(block: false, alt: "70 degrees")[$70^(∘)$] and #math.equation(block: false, alt: "430 degrees")[$430^(∘)$] have the same terminal side because #math.equation(block: false, alt: "430 degrees equals 70 degrees plus 360 degrees")[$430^(∘) = 70^(∘) + 360^(∘)$]. Such angles are called #strong[coterminal].


        The angle #math.equation(block: false, alt: "790 degrees")[$790^(∘)$] is also coterminal with #math.equation(block: false, alt: "70 degrees")[$70^(∘)$], because if we add two revolutions to #math.equation(block: false, alt: "70 degrees")[$70^(∘)$], we get #math.equation(block: false, alt: "790 degrees equals 70 degrees plus 2 open parenthesis 360 degrees close parenthesis")[$" " 790^(∘) = 70^(∘) + 2 ( 360^(∘) )$], as shown below.


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


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


        Because coterminal angles have the same standard position, their trigonometric ratios are equal. For example, you can verify that, to four decimal places,


        #math.equation(block: true, alt: "cos ⁡ open parenthesis 790 degrees close parenthesis equals cos ⁡ open parenthesis 70 degrees close parenthesis equals 0.3420")[$cos ( 790^(∘) ) = cos ( 70^(∘) ) = 0.3420$]
        #examplebox("Example 8")[][
+ Find an angle between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] that is coterminal with #math.equation(block: false, alt: "520 degrees")[$520^(∘)$].
+ Use a calculator to verify that the trig ratios of #math.equation(block: false, alt: "520 degrees")[$520^(∘)$] are the same as the ratios of the coterminal angle.


          
          #solutionbox[
+ We subtract multiples of #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] from #math.equation(block: false, alt: "520 degrees")[$520^(∘)$] until the remainder is less than #math.equation(block: false, alt: "360 degrees")[$360^(∘)$].
        #math.equation(block: true, alt: "520 degrees minus 360 degrees equals 160 degrees")[$520^(∘) − 360^(∘) = 160^(∘)$]
    Because #math.equation(block: false, alt: "160 degrees")[$160^(∘)$] is between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$], this is the angle we want.
+ You can use your calculator  to see that
        #math.equation(block: true, alt: "sin ⁡ open parenthesis 160 degrees close parenthesis, equals sin ⁡ open parenthesis 520 degrees close parenthesis equals 0.3420; cos ⁡ open parenthesis 160 degrees close parenthesis, equals cos ⁡ open parenthesis 520 degrees close parenthesis equals minus 0.9397; tan ⁡ open parenthesis 160 degrees close parenthesis, equals tan ⁡ open parenthesis 520 degrees close parenthesis equals minus 0.3640")[$sin ( 160^(∘) ) & = sin ( 520^(∘) ) = 0.3420 \ cos ( 160^(∘) ) & = cos ( 520^(∘) ) = − 0.9397 \ tan ( 160^(∘) ) & = tan ( 520^(∘) ) = − 0.3640$]
]
]


        If the direction of rotation  is important, we let positive angles represent rotation in the counter-clockwise direction, and negative angles represent rotation in the clockwise direction. For example, the angle #math.equation(block: false, alt: "minus 60 degrees")[$− 60^(∘)$] shown at right lies in the fourth quadrant. It is coterminal with  #math.equation(block: false, alt: "minus 60 degrees plus 360 degrees equals 300 degrees")[$− 60^(∘) + 360^(∘) = 300^(∘)$].


        #figure(figph[positive and negative angle], alt: "positive and negative angle", caption: none)


        
          Find two angles coterminal with #math.equation(block: false, alt: "102 degrees")[$102^(∘)$], one positive and one negative.


          
            Add #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] to get #math.equation(block: false, alt: "462 degrees")[$462^(∘)$], and subtract #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] to get #math.equation(block: false, alt: "minus 258 degrees")[$attach(− 258, t: ∘)$]


          
        
      
      Review the following skills you will need for this section.


      #notebox("Algebra Refresher", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate the function.


        + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 2 x")[$f ( x ) = x^(2) − 2 x$]+ #math.equation(block: false, alt: "f open parenthesis minus 3 close parenthesis")[$f ( − 3 )$]
+ #math.equation(block: false, alt: "f open parenthesis a minus 3 close parenthesis")[$f ( a − 3 )$]
+ #math.equation(block: false, alt: "f open parenthesis a close parenthesis minus 5")[$f ( a ) − 5$]
+ #math.equation(block: false, alt: "f open parenthesis a close parenthesis minus f open parenthesis 5 close parenthesis")[$f ( a ) − f ( 5 )$]
+ #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of x plus 4")[$g ( x ) = sqrt(x + 4)$]+ #math.equation(block: false, alt: "g open parenthesis 9 close parenthesis")[$g ( 9 )$]
+ #math.equation(block: false, alt: "g open parenthesis 4 h close parenthesis")[$g ( 4 h )$]
+ #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis plus g open parenthesis 1 close parenthesis")[$g ( 0 ) + g ( 1 )$]
+ #math.equation(block: false, alt: "g open parenthesis c squared close parenthesis")[$g ( c^(2) )$]
+ #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals the fraction 2 over x")[$F ( x ) = display(frac(2, x))$]+ #math.equation(block: false, alt: "F open parenthesis the fraction minus 1 over 2 close parenthesis")[$F ( display(frac(− 1, 2)) )$]
+ #math.equation(block: false, alt: "F open parenthesis the fraction w over 2 close parenthesis")[$F ( display(frac(w, 2)) )$]
+ #math.equation(block: false, alt: "F open parenthesis w plus 2 close parenthesis")[$F ( w + 2 )$]
+ #math.equation(block: false, alt: "F open parenthesis w close parenthesis plus F open parenthesis 2 close parenthesis")[$F ( w ) + F ( 2 )$]
+ #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals 2 to the power x")[$G ( x ) = 2^(x)$]+ #math.equation(block: false, alt: "G open parenthesis minus 3 close parenthesis")[$G ( − 3 )$]
+ #math.equation(block: false, alt: "G open parenthesis a plus 3 close parenthesis")[$G ( a + 3 )$]
+ #math.equation(block: false, alt: "G open parenthesis a close parenthesis plus G open parenthesis 3 close parenthesis")[$G ( a ) + G ( 3 )$]
+ #math.equation(block: false, alt: "G open parenthesis the fraction 3 over 2 close parenthesis")[$G ( display(frac(3, 2)) )$]


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


        Algebra Refresher Answers


        + + #math.equation(block: false, alt: "15")[$15$]
+ #math.equation(block: false, alt: "a squared minus 8 a plus 15")[$a^(2) − 8 a + 15$]
+ #math.equation(block: false, alt: "a squared minus 2 a minus 5")[$a^(2) − 2 a − 5$]
+ #math.equation(block: false, alt: "a squared minus 2 a minus 15")[$a^(2) − 2 a − 15$]
+ + #math.equation(block: false, alt: "the square root of 13")[$sqrt(13)$]
+ #math.equation(block: false, alt: "2 the square root of h plus 1")[$2 sqrt(h + 1)$]
+ #math.equation(block: false, alt: "2 plus the square root of 5")[$2 + sqrt(5)$]
+ #math.equation(block: false, alt: "the square root of c squared plus 4")[$sqrt(c^(2) + 4)$]
+ + #math.equation(block: false, alt: "minus 4")[$− 4$]
+ #math.equation(block: false, alt: "the fraction 4 over w")[$display(frac(4, w))$]
+ #math.equation(block: false, alt: "the fraction 2 over w plus 2")[$display(frac(2, w + 2))$]
+ #math.equation(block: false, alt: "the fraction 2 over w plus 1")[$display(frac(2, w)) + 1$]
+ + #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$]
+ #math.equation(block: false, alt: "8 open parenthesis 2 to the power a close parenthesis")[$8 ( 2^(a) )$]
+ #math.equation(block: false, alt: "2 to the power a plus 8")[$2^(a) + 8$]
+ #math.equation(block: false, alt: "2 the square root of 2")[$2 sqrt(2)$]
]


    
    
      
=== Section 4.1 Summary


      
        
==== Vocabulary


        - Standard position
- Reference angle
- Reference triangle
- Coterminal angle


      
      
        
==== 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 ⁡ open parenthesis θ close parenthesis equals the fraction y over r cos ⁡ open parenthesis θ close parenthesis equals the fraction x over r tan ⁡ open parenthesis θ close parenthesis 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[angles], alt: "angles", 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.


      
      
        
====  Study Questions


        + Explain why #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis less than or equal to 1")[$cos ( θ ) ≤ 1$] for any angle #math.equation(block: false, alt: "θ")[$θ$].
+ Is it true that #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis less than or equal to 1")[$tan ( θ ) ≤ 1$] for any angle #math.equation(block: false, alt: "θ")[$θ$] ?  Explain.
+ Sketch a figure showing how to compute the reference angle for angles in each of the four quadrants.
+ True or false:  If #math.equation(block: false, alt: "β greater than α")[$β > α$], then #math.equation(block: false, alt: "sin ⁡ open parenthesis β close parenthesis greater than sin ⁡ α")[$sin ( β ) > sin α$].
+ How many angles have a cosine equal to 0.4? How many angles between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] have a cosine equal to 0.4?


      
      
        
==== Skills


        + Use angles to represent rotations    \#1-6
+ Sketch angles in standard position    \#7-12
+ Find coterminal angles    \#13-24
+ Find and use reference angles    \#25-44, 55-64
+ Find trigonometric ratios for the special angles    \#45-54


      
    
    
      
=== Homework 4.1


      
        How many degrees are in each angle?


        + #math.equation(block: false, alt: "the fraction 3 over 5")[$display(frac(3, 5))$] of one rotation
+ #math.equation(block: false, alt: "the fraction 3 over 10")[$display(frac(3, 10))$] of one rotation
+ #math.equation(block: false, alt: "the fraction 4 over 3")[$display(frac(4, 3))$] of one rotation
+ #math.equation(block: false, alt: "the fraction 8 over 3")[$display(frac(8, 3))$] of one rotation


        
          + #math.equation(block: false, alt: "216 degrees")[$216^(∘)$]
+ #math.equation(block: false, alt: "108 degrees")[$108^(∘)$]
+ #math.equation(block: false, alt: "480 degrees")[$480^(∘)$]
+ #math.equation(block: false, alt: "960 degrees")[$960^(∘)$]


        
      
      
        How many degrees are in each angle?


        + #math.equation(block: false, alt: "the fraction 5 over 6")[$display(frac(5, 6))$] of one rotation
+ #math.equation(block: false, alt: "the fraction 3 over 8")[$display(frac(3, 8))$] of one rotation
+ #math.equation(block: false, alt: "the fraction 7 over 4")[$display(frac(7, 4))$] of one rotation
+ #math.equation(block: false, alt: "the fraction 7 over 12")[$display(frac(7, 12))$] of one rotation


      
      
        What fraction of a complete rotation is represented by each angle?


        + #math.equation(block: false, alt: "45 degrees")[$45^(∘)$]
+ #math.equation(block: false, alt: "300 degrees")[$300^(∘)$]
+ #math.equation(block: false, alt: "540 degrees")[$540^(∘)$]
+ #math.equation(block: false, alt: "420 degrees")[$420^(∘)$]


        
          + #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$]
+ #math.equation(block: false, alt: "the fraction 5 over 6")[$display(frac(5, 6))$]
+ #math.equation(block: false, alt: "the fraction 3 over 2")[$display(frac(3, 2))$]
+ #math.equation(block: false, alt: "the fraction 7 over 6")[$display(frac(7, 6))$]


        
      
      
        What fraction of a complete rotation is represented by each angle?


        + #math.equation(block: false, alt: "60 degrees")[$60^(∘)$]
+ #math.equation(block: false, alt: "240 degrees")[$240^(∘)$]
+ #math.equation(block: false, alt: "450 degrees")[$450^(∘)$]
+ #math.equation(block: false, alt: "150 degrees")[$150^(∘)$]


      
      
        + Through what angle does the hour hand of a clock rotate between 2 pm and 10 pm?
+ Through what angle does the hour hand of a clock rotate between 2 am and 10 pm?


        
          + #math.equation(block: false, alt: "the fraction 2 over 3")[$display(frac(2, 3))$]
+ #math.equation(block: false, alt: "the fraction 5 over 3")[$display(frac(5, 3))$]


        
      
      
        + Through what angle does the minute hand of a clock rotate between 3:25 am and 3:50 am?
+ Through what angle does the minute hand of a clock rotate between 4:10 pm and 6:25 pm?


      
      For Problems 7–12, calculate the degree measure of the unknown angle, and sketch the angle in standard position.


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


        
          #math.equation(block: true, alt: "60 degrees")[$60^(∘)$]


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


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


        
          #math.equation(block: true, alt: "60 degrees")[$60^(∘)$]


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


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


        
          #math.equation(block: true, alt: "14 degrees")[$14^(∘)$]


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


      
      For Problems 13–18, find two angles, one positive and one negative, that are coterminal with the given angle.


      
        #math.equation(block: true, alt: "40 degrees")[$40^(∘)$]


        
          #math.equation(block: false, alt: "400 degrees")[$400^(∘)$] and #math.equation(block: false, alt: "minus 320 degrees")[$− 320^(∘)$] (Answers vary.)


        
      
      
        #math.equation(block: true, alt: "160 degrees")[$160^(∘)$]


      
      
        #math.equation(block: true, alt: "215 degrees")[$215^(∘)$]


        
          #math.equation(block: false, alt: "575 degrees")[$575^(∘)$] and #math.equation(block: false, alt: "minus 145 degrees")[$− 145^(∘)$] (Answers vary.)


        
      
      
        #math.equation(block: true, alt: "250 degrees")[$250^(∘)$]


      
      
        #math.equation(block: true, alt: "305 degrees")[$305^(∘)$]


        
          #math.equation(block: false, alt: "665 degrees")[$665^(∘)$] and #math.equation(block: false, alt: "minus 55 degrees")[$− 55^(∘)$] (Answers vary.)


        
      
      
        #math.equation(block: true, alt: "340 degrees")[$340^(∘)$]


      
      For Problems 19–24, find a positive angle between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] that is coterminal with the given angle.


      
        #math.equation(block: true, alt: "minus 65 degrees")[$− 65^(∘)$]


        
          #math.equation(block: true, alt: "295 degrees")[$295^(∘)$]


        
      
      
        #math.equation(block: true, alt: "minus 140 degrees")[$− 140^(∘)$]


      
      
        #math.equation(block: true, alt: "minus 290 degrees")[$− 290^(∘)$]


        
          #math.equation(block: true, alt: "70 degrees")[$70^(∘)$]


        
      
      
        #math.equation(block: true, alt: "minus 325 degrees")[$− 325^(∘)$]


      
      
        #math.equation(block: true, alt: "minus 405 degrees")[$− 405^(∘)$]


        
          #math.equation(block: true, alt: "315 degrees")[$315^(∘)$]


        
      
      
        #math.equation(block: true, alt: "minus 750 degrees")[$− 750^(∘)$]


      
      For Problems 25–26, use the grid provided below.


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


      
        Draw two different angles #math.equation(block: false, alt: "α")[$α$] and #math.equation(block: false, alt: "β")[$β$] in standard position whose sine is #math.equation(block: false, alt: "0.6")[$0.6$]. Note that the radius of the circle is 1,


        + Use a protractor to measure #math.equation(block: false, alt: "α")[$α$] and #math.equation(block: false, alt: "β")[$β$].
+ Find the reference angles for both #math.equation(block: false, alt: "α")[$α$] and #math.equation(block: false, alt: "β")[$β$]. Draw in the reference triangles.


        
          + #math.equation(block: false, alt: "36.9 degrees , 143.1 degrees")[$36.9^(∘) , " " 143.1^(∘)$]
+ #figure(figph[angles on grid], alt: "angles on grid", caption: none)


        
      
      
        Draw two different angles #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "ϕ")[$ϕ$] in standard position whose sine is #math.equation(block: false, alt: "minus 0.8")[$− 0.8$].


        + Use a protractor to measure #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "ϕ")[$ϕ$].
+ Find the reference angles for both #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "ϕ")[$ϕ$]. Draw in the reference triangles.


      
      For Problems 27–28, use the grid provided below.


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


      
        Draw two different angles #math.equation(block: false, alt: "α")[$α$] and #math.equation(block: false, alt: "β")[$β$] in standard position whose cosine is #math.equation(block: false, alt: "0.3")[$0.3$].


        + Use a protractor to measure #math.equation(block: false, alt: "α")[$α$] and #math.equation(block: false, alt: "β")[$β$].
+ Find the reference angles for both #math.equation(block: false, alt: "α")[$α$] and #math.equation(block: false, alt: "β")[$β$]. Draw in the reference triangles.


        
          + #math.equation(block: false, alt: "72.5 degrees , 287.5 degrees")[$72.5^(∘) , " " 287.5^(∘)$]
+ #figure(figph[graph], alt: "graph", caption: none)


        
      
      
        Draw two different angles #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "ϕ")[$ϕ$] in standard position whose cosine is #math.equation(block: false, alt: "minus 0.4")[$− 0.4$].


        + Use a protractor to measure #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "ϕ")[$ϕ$].
+ Find the reference angles for both #math.equation(block: false, alt: "θ")[$θ$] and #math.equation(block: false, alt: "ϕ")[$ϕ$]. Draw in the reference triangles.


      
      For Problems 29–34, find the reference angle. Make a sketch showing the angle, the reference angle, and the reference triangle.


      
        #math.equation(block: true, alt: "100 degrees")[$100^(∘)$]


        
          #math.equation(block: true, alt: "80 degrees")[$80^(∘)$]


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


        
      
      
        #math.equation(block: true, alt: "125 degrees")[$125^(∘)$]


      
      
        #math.equation(block: true, alt: "216 degrees")[$216^(∘)$]


        
          #math.equation(block: true, alt: "36 degrees")[$36^(∘)$]


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


        
      
      
        #math.equation(block: true, alt: "242 degrees")[$242^(∘)$]


      
      
        #math.equation(block: true, alt: "297 degrees")[$297^(∘)$]


        
          #math.equation(block: true, alt: "63 degrees")[$63^(∘)$]


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


        
      
      
        #math.equation(block: true, alt: "336 degrees")[$336^(∘)$]


      
      For Problems 35–40, find three angles between #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] with the given reference angle, and sketch all four angles on the same grid.


      
        #math.equation(block: true, alt: "15 degrees")[$15^(∘)$]


        
          #math.equation(block: true, alt: "165 degrees")[$165^(∘)$], #math.equation(block: true, alt: "95 degrees")[$95^(∘)$], #math.equation(block: true, alt: "345 degrees")[$345^(∘)$]


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


        
      
      
        #math.equation(block: true, alt: "26 degrees")[$26^(∘)$]


      
      
        #math.equation(block: true, alt: "40 degrees")[$40^(∘)$]


        
          #math.equation(block: true, alt: "140 degrees")[$140^(∘)$], #math.equation(block: true, alt: "220 degrees")[$220^(∘)$], #math.equation(block: true, alt: "320 degrees")[$320^(∘)$]


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


        
      
      
        #math.equation(block: true, alt: "50 degrees")[$50^(∘)$]


      
      
        #math.equation(block: true, alt: "68 degrees")[$68^(∘)$]


        
          #math.equation(block: true, alt: "112 degrees")[$112^(∘)$], #math.equation(block: true, alt: "248 degrees")[$248^(∘)$], #math.equation(block: true, alt: "292 degrees")[$292^(∘)$]


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


        
      
      
        #math.equation(block: true, alt: "75 degrees")[$75^(∘)$]


      
      For Problems 41–48, use the values given below to find the trigonometric ratio.  Do not use a calculator!


      #math.equation(block: true, alt: "cos ⁡ open parenthesis 23 degrees close parenthesis equals 0.9205 sin ⁡ open parenthesis 46 degrees close parenthesis equals 0.7193 tan ⁡ open parenthesis 78 degrees close parenthesis equals 4.7046")[$cos ( 23^(∘) ) = 0.9205 " " " " " " " " " " " " sin ( 46^(∘) ) = 0.7193 " " " " " " " " " " " " tan ( 78^(∘) ) = 4.7046$]
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis 157 degrees close parenthesis")[$cos ( 157^(∘) )$]


        
          #math.equation(block: true, alt: "minus 0.9205")[$− 0.9205$]


        
      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis 226 degrees close parenthesis")[$sin ( 226^(∘) )$]


      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis 314 degrees close parenthesis")[$sin ( 314^(∘) )$]


        
          #math.equation(block: true, alt: "minus 0.7193")[$− 0.7193$]


        
      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis 203 degrees close parenthesis")[$cos ( 203^(∘) )$]


      
      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis 258 degrees close parenthesis")[$tan ( 258^(∘) )$]


        
          #math.equation(block: true, alt: "4.705")[$4.705$]


        
      
      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis 282 degrees close parenthesis")[$tan ( 282^(∘) )$]


      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis minus 134 degrees close parenthesis")[$sin ( − 134^(∘) )$]


        
          #math.equation(block: true, alt: "minus 0.7193")[$− 0.7193$]


        
      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis minus 383 degrees close parenthesis")[$cos ( − 383^(∘) )$]


      
      
        On the circle in the figure, all angles are shown in standard position. Find the measure in degrees of the angles labeled (a)-(i).


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


        
          + #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: "210 degrees")[$210^(∘)$]
+ #math.equation(block: false, alt: "225 degrees")[$225^(∘)$]
+ #math.equation(block: false, alt: "240 degrees")[$240^(∘)$]
+ #math.equation(block: false, alt: "300 degrees")[$300^(∘)$]
+ #math.equation(block: false, alt: "315 degrees")[$315^(∘)$]
+ #math.equation(block: false, alt: "330 degrees")[$330^(∘)$]


        
      
      
        Find the reference angle for each of your answers in Problem 45.


      
      
        + Draw three angles, one in each quadrant except the first, whose reference angle is #math.equation(block: false, alt: "60 degrees")[$60^(∘)$].
+ Find exact values for the sine, cosine, and tangent of each of the angles in part (a).


        
          + #figure(figph[angles], alt: "angles", caption: none)
+ #math.equation(block: false, alt: "sin ⁡ open parenthesis 120 degrees close parenthesis equals the fraction the square root of 3 over 2 , cos ⁡ open parenthesis 120 degrees close parenthesis equals the fraction minus 1 over 2 , tan ⁡ open parenthesis 120 degrees close parenthesis equals minus the square root of 3 ,")[$sin ( 120^(∘) ) = display(frac(sqrt(3), 2)) , " " cos ( 120^(∘) ) = display(frac(− 1, 2)) , " " tan ( 120^(∘) ) = − sqrt(3) ,$] #linebreak() #math.equation(block: false, alt: "sin ⁡ open parenthesis 240 degrees close parenthesis equals the fraction minus the square root of 3 over 2 , cos ⁡ open parenthesis 240 degrees close parenthesis equals the fraction minus 1 over 2 , tan ⁡ open parenthesis 240 degrees close parenthesis equals the square root of 3 ,")[$sin ( 240^(∘) ) = display(frac(− sqrt(3), 2)) , " " cos ( 240^(∘) ) = display(frac(− 1, 2)) , " " tan ( 240^(∘) ) = sqrt(3) ,$] #linebreak() #math.equation(block: false, alt: "sin ⁡ open parenthesis 300 degrees close parenthesis equals the fraction minus the square root of 3 over 2 , cos ⁡ open parenthesis 300 degrees close parenthesis equals the fraction 1 over 2 , tan ⁡ open parenthesis 300 degrees close parenthesis equals minus the square root of 3")[$sin ( 300^(∘) ) = display(frac(− sqrt(3), 2)) , " " cos ( 300^(∘) ) = display(frac(1, 2)) , " " tan ( 300^(∘) ) = − sqrt(3)$]


        
      
      
        + Draw three angles, one in each quadrant except the first, whose reference angle is #math.equation(block: false, alt: "30 degrees")[$30^(∘)$].
+ Find exact values for the sine, cosine, and tangent of each of the angles in part (a).


      
      
        + Draw three angles, one in each quadrant except the first, whose reference angle is #math.equation(block: false, alt: "45 degrees")[$45^(∘)$].
+ Find exact values for the sine, cosine, and tangent of each of the angles in part (a).


        
          + #figure(figph[angles], alt: "angles", caption: none)
+ #math.equation(block: false, alt: "sin ⁡ open parenthesis 135 degrees close parenthesis equals the fraction 1 over the square root of 2 , cos ⁡ open parenthesis 135 degrees close parenthesis equals the fraction minus 1 over the square root of 2 , open parenthesis tan ⁡ 135 degrees close parenthesis equals minus 1 ,")[$sin ( 135^(∘) ) = display(frac(1, sqrt(2))) , " " cos ( 135^(∘) ) = display(frac(− 1, sqrt(2))) , " " ( tan 135^(∘) ) = − 1 ,$] #linebreak() #math.equation(block: false, alt: "sin ⁡ open parenthesis 225 degrees close parenthesis equals the fraction minus 1 over the square root of 2 , cos ⁡ open parenthesis 225 degrees close parenthesis equals the fraction minus 1 over the square root of 2 , tan ⁡ open parenthesis 225 degrees close parenthesis equals 1 ,")[$sin ( 225^(∘) ) = display(frac(− 1, sqrt(2))) , " " cos ( 225^(∘) ) = display(frac(− 1, sqrt(2))) , " " tan ( 225^(∘) ) = 1 ,$] #linebreak() #math.equation(block: false, alt: "sin ⁡ open parenthesis 315 degrees close parenthesis equals the fraction minus 1 over the square root of 2 , cos ⁡ open parenthesis 315 degrees close parenthesis equals the fraction 1 over the square root of 2 , tan ⁡ open parenthesis 315 degrees close parenthesis equals minus 1")[$sin ( 315^(∘) ) = display(frac(− 1, sqrt(2))) , " " cos ( 315^(∘) ) = display(frac(1, sqrt(2))) , " " tan ( 315^(∘) ) = − 1$]


        
      
      
        Complete the table with exact values.


        #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: "45 degrees")[$45^(∘)$]], [#math.equation(block: false, alt: "60 degrees")[$60^(∘)$]], [#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: "210 degrees")[$210^(∘)$]], [#math.equation(block: false, alt: "225 degrees")[$225^(∘)$]], [#math.equation(block: false, alt: "240 degrees")[$240^(∘)$]], [#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: "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($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [#math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis")[$sin ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
  [#math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis")[$tan ( θ )$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$],
))


      
      
        In which two quadrants is the statement true?


        + The sine is negative.
+ The cosine is negative.
+ The tangent is positive.


        
          + III and IV
+ II and III
+ I and III


        
      
      
        Find all angles between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] for which the statement is true.


        + #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis equals minus 1")[$cos ( θ ) = − 1$]
+ #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis equals minus 1")[$sin ( θ ) = − 1$]
+ #math.equation(block: false, alt: "tan ⁡ open parenthesis θ close parenthesis equals minus 1")[$tan ( θ ) = − 1$]


      
      
        + Find two angles, #math.equation(block: false, alt: "0 less than or equal to θ less than 360 degrees")[$0 ≤ θ < 360^(∘)$], with #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis equals 0")[$sin ( θ ) = 0$].
+ Find two angles, #math.equation(block: false, alt: "0 less than or equal to θ less than 360 degrees")[$0 ≤ θ < 360^(∘)$], with #math.equation(block: false, alt: "cos ⁡ open parenthesis θ close parenthesis equals 0")[$cos ( θ ) = 0$].


        
          + #math.equation(block: false, alt: "0 degrees and 180 degrees")[$0^(∘) " " "and" " " 180^(∘)$]
+ #math.equation(block: false, alt: "90 degrees and 270 degrees")[$90^(∘) " " "and" " " 270^(∘)$]


        
      
      
        + Find two angles, #math.equation(block: false, alt: "0 less than or equal to θ less than 360 degrees")[$0 ≤ θ < 360^(∘)$], with #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis equals cos ⁡ open parenthesis θ close parenthesis")[$sin ( θ ) = cos ( θ )$].
+ Find two angles, #math.equation(block: false, alt: "0 less than or equal to θ less than 360 degrees")[$0 ≤ θ < 360^(∘)$], with #math.equation(block: false, alt: "sin ⁡ open parenthesis θ close parenthesis equals minus cos ⁡ open parenthesis θ close parenthesis")[$sin ( θ ) = − cos ( θ )$].


      
      For Problems 59–64, find a second angle between #math.equation(block: false, alt: "0 degrees")[$0^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] with the given trigonometric ratio.


      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis 75 degrees close parenthesis")[$sin ( 75^(∘) )$]


        
          #math.equation(block: true, alt: "105 degrees")[$105^(∘)$]


        
      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis 32 degrees close parenthesis")[$cos ( 32^(∘) )$]


      
      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis 84 degrees close parenthesis")[$tan ( 84^(∘) )$]


        
          #math.equation(block: true, alt: "264 degrees")[$264^(∘)$]


        
      
      
        #math.equation(block: true, alt: "sin ⁡ open parenthesis 16 degrees close parenthesis")[$sin ( 16^(∘) )$]


      
      
        #math.equation(block: true, alt: "cos ⁡ open parenthesis 47 degrees close parenthesis")[$cos ( 47^(∘) )$]


        
          #math.equation(block: true, alt: "313 degrees")[$313^(∘)$]


        
      
      
        #math.equation(block: true, alt: "tan ⁡ open parenthesis 56 degrees close parenthesis")[$tan ( 56^(∘) )$]


      
      
        Explain why the definitions of the trigonometric ratios for a third-quadrant angle (between #math.equation(block: false, alt: "180 degrees")[$180^(∘)$] and #math.equation(block: false, alt: "270 degrees")[$270^(∘)$]) are independent of the point #math.equation(block: false, alt: "P")[$P$] chosen on the terminal side. Illustrate with a figure.


        
          Sides of similar triangles are proportional.


        
      
      
        Explain why the definitions of the trigonometric ratios for a fourth-quadrant angle (between #math.equation(block: false, alt: "270 degrees")[$270^(∘)$] and #math.equation(block: false, alt: "360 degrees")[$360^(∘)$]) are independent of the point #math.equation(block: false, alt: "P")[$P$] chosen on the terminal side. Illustrate with a figure.