6. Trigonometric Identities and More Equations

Given $\tan\theta=\frac{5}{12}$ and 0 < $\theta$ < $\frac{\pi}{2}$, find $\cos\left(2\theta\right)$.

Find values of the sine and cosine functions for each angle measure.

2x, given tan x = 5/3  and sin x < 0

Match each expression in Column I with its value in Column II.

10. cos 67.5°

Find values of the sine and cosine functions for each angle measure.

2θ, given cos θ = (√3)/5  and sin θ > 0

Use a half-angle identity to find each exact value.

sin 195°

Find values of the sine and cosine functions for each angle measure.

θ, given cos 2θ = 3/4 and θ terminates in quadrant III

Use a half-angle identity to find each exact value.

cos 195°

Find values of the sine and cosine functions for each angle measure.

θ, given cos 2θ = 2/3 and 90° < θ <180°

Use a half-angle identity to find each exact value.

sin 165°

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.)

Use the given information to find each of the following.

sin x/2 , given cos x = - 5/8, with π/2 < x < π

Match each expression in Column I with its value in Column II.

(2 tan 15°)/(1 - tan² 15°)

Determine whether the positive or negative square root should be selected.

cos 58° = ±√ (1 + cos 116°)/2]

Use the given information to find each of the following.

cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°

Use the given information to find each of the following.

cos x/2 , given cot x = -3, with π/2 < x < π

Use the given information to find each of the following.

cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°

Use the given information to find each of the following.

cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II

Use the given information to find each of the following.

sin x, given cos 2x = 2/3 , with π < x < 3π/2

If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

Simplify each expression.

√[(1 + cos 76°)/2]

Simplify each expression.

√[(1 + cos 165°)/(1 - cos 165°)]

Simplify each expression. See Example 4.

2 tan 15°/(1 - tan² 15°)

Simplify each expression.

sin 158.2°/(1 + cos 158.2°)

Simplify each expression.

±√[(1 + cos 18x)/2]

Match each expression in Column I with its value in Column II.

cos² (π/6) - sin² (π/6)

Simplify each expression. See Example 4.

1 - 2 sin² 22 ½°

Determine whether the positive or negative square root should be selected.

sin (-10°) = ± √[(1 - cos (-20°))/2]

Simplify each expression.

±√[(1 + cos 20α)/2]

Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

Simplify each expression. See Example 4.

cos² π/8 - 1/2

Simplify each expression.

±√[(1 - cos 5A)/(1 + cos 5A)]

Simplify each expression.

± √[(1 + cos (x/4))/2]

Simplify each expression. See Example 4.

tan 34°/2(1 - tan² 34°)

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(1 - cos 2x)/sin 2x

Simplify each expression.

±√[(1 - cos (3θ/5))/2]

Simplify each expression. See Example 4.

⅛ sin 29.5° cos 29.5°

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(cos x sin 2x)/1 + cos 2x)

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Simplify each expression. See Example 4.

cos² 2x - sin² 2x

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

csc x - cot x

Verify that each equation is an identity.

(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

Express each function as a trigonometric function of x. See Example 5.

cos 3x

Verify that each equation is an identity.

tan (θ/2) = csc θ - cot θ

Express each function as a trigonometric function of x. See Example 5.

cos 4x

Verify that each equation is an identity.

cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

Write each expression as a sum or difference of trigonometric functions. See Example 7.

2 cos 85° sin 140°

Match each expression in Column I with its value in Column II.

(2 tan (π/3))/(1 - tan² (π/3))

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.

cos 18°

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.

cot 18°

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°

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.

tan 72°

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 72°

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.

sin 162°

Find values of the sine and cosine functions for each angle measure.

2θ, given cos θ = -12/13 and sin θ > 0

Match each expression in Column I with its value in Column II.

8. tan (-π/8)