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Problem 52
Express each function as a trigonometric function of x. See Example 5.
cos 4x
Problem 58
Write each expression as a sum or difference of trigonometric functions. See Example 7.
2 cos 85° sin 140°
Problem 60
Write each expression as a sum or difference of trigonometric functions. See Example 7.
5 cos 3x cos 2x
Problem 62
Write each expression as a sum or difference of trigonometric functions. See Example 7.
8 sin 7x sin 9x
Problem 64
Write each expression as a product of trigonometric functions. See Example 8.
cos 5x + cos 8x
Problem 66
Write each expression as a product of trigonometric functions. See Example 8.
sin 102° - sin 95°
Problem 68
Write each expression as a product of trigonometric functions. See Example 8.
sin 9x - sin 3x
Problem 5.43
Simplify each expression.
± √[(1 + cos (x/4))/2]
Problem 5.41
Simplify each expression.
±√[(1 - cos 8θ)/(1 + cos 8θ)]
Problem 5.6.2
Determine whether the positive or negative square root should be selected.
cos 58° = ±√ (1 + cos 116°)/2]
Problem 5.6.12
Use a half-angle identity to find each exact value.
sin 195°
Problem 5.6.10
Match each expression in Column I with its value in Column II.
10. cos 67.5°
Problem 5.6.18
The half-angle identity
tan A/2 = ± √[(1 - cosA)/(1 + cos A)]
can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity
tan A/2 = sin A/(1 + cos A)
can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)
Problem 5.69
Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
csc 18°
Problem 5.6.8
Match each expression in Column I with its value in Column II.
8. tan (-π/8)
Problem 5.6.4
Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]
Problem 14
Use a half-angle identity to find each exact value.
cos 195°
Problem 16
Use a half-angle identity to find each exact value.
sin 165°
Problem 20
Use the given information to find each of the following.
sin x/2 , given cos x = - 5/8, with π/2 < x < π
Problem 22
Use the given information to find each of the following.
cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°
Problem 24
Use the given information to find each of the following.
cos x/2 , given cot x = -3, with π/2 < x < π
Problem 26
Use the given information to find each of the following.
cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°
Problem 28
Use the given information to find each of the following.
cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II
Problem 30
Use the given information to find each of the following.
sin x, given cos 2x = 2/3 , with π < x < 3π/2
Problem 32
If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .
Problem 34
Simplify each expression.
√[(1 + cos 76°)/2]
Problem 36
Simplify each expression.
√[(1 + cos 165°)/(1 - cos 165°)]
Problem 38
Simplify each expression.
sin 158.2°/(1 + cos 158.2°)
Problem 39
Simplify each expression.
±√[(1 + cos 18x)/2]
Problem 40
Simplify each expression.
±√[(1 + cos 20α)/2]