404 CHAPTER 6 Trigonometric Functions 5 Find the Exact Values of the Trigonometric Functions for Integer Multiples of π π = ° = ° 6 30 , 4 45 , and π = ° 3 60 We know the exact values of the trigonometric functions of 4 45 . π = ° Using symmetry, we can find the exact values of the trigonometric functions of π π = ° = ° 3 4 135 , 5 4 225 , and 7 4 315 . π = ° The point on the unit circle corresponding to 4 45 π = ° is 2 2 , 2 2 . ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ See Figure 30. Using symmetry with respect to the y-axis, the point 2 2 , 2 2 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ is the point on the unit circle that corresponds to the angle 3 4 135 . π = ° Similarly, using symmetry with respect to the origin, the point 2 2 , 2 2 − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ is the point on the unit circle that corresponds to the angle 5 4 225 . π = ° Finally, using symmetry with respect to the x-axis, the point 2 2 , 2 2 − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ is the point on the unit circle that corresponds to the angle 7 4 315 . π = ° Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, this length is bent up at an angle .θ See Figure 29. The area A of the opening may be expressed as a function of θ as θ θ θ ( ) ( ) = + A 16 sin cos 1 Find the area A of the opening for 30 , 45 , θ θ = ° = ° and 60 . θ = ° EXAMPLE 8 Solution θ ( ) ( ) = ° ° = ° ° + = ⋅ ⋅ + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + ≈ A For 30 : 30 16 sin30 cos30 1 16 1 2 3 2 1 4 3 8 14.93 The area of the opening for 30 θ = ° is about 14.93 square inches. θ ( ) ( ) = ° ° = ° ° + = ⋅ ⋅ + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + ≈ A For 45 : 45 16 sin45 cos45 1 16 2 2 2 2 1 8 8 2 19.31 The area of the opening for 45 θ = ° is about 19.31 square inches. θ ( ) ( ) ( ) = ° ° = ° ° + = ⋅ ⋅ + = ≈ A For 60 : 60 16 sin60 cos60 1 16 3 2 1 2 1 12 3 20.78 The area of the opening for 60 θ = ° is about 20.78 square inches. Figure 29 u u 4 in. 12 in. 4 in. 4 in. 4 in. 4 in. Figure 30 p–– 4 1 21 21 1 y x 2 –– 2 2 –– 2 2 , ) ( 2 2 2 ( 2 –– 2 2 –– 2 ( , ) 2 –– 2 2 –– 2 ) , 2 –– 2 2 –– 2 ) ( , 5p––– 4 7p––– 4 3p––– 4 Finding Exact Values for Multiples of π = ° 4 45 Find the exact value of each expression. (a) cos 5 4 π (b) sin135° (c) tan315° (d) sin 4 π ( ) − (e) cos 11 4 π EXAMPLE 9
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