6. Trigonometric Identities and More Equations

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

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