Simplify each expression.

±√[(1 + cos 20α)/2]

Simplify each expression.

±√[(1 - cos 5A)/(1 + cos 5A)]

Simplify each expression.

±√[(1 - cos (3θ/5))/2]

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Verify that each equation is an identity.

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

Verify that each equation is an identity.

tan (θ/2) = csc θ - cot θ

Verify that each equation is an identity.

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

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.

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°

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

For each expression in Column I, choose the expression from Column II that completes an identity.

6. sec² x = ____

II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

Use the given information to find sin(x + y).

cos x = 2/9, sin y = - 1, x in quadrant IV, y in quadrant III

Use the given information to find the quadrant of x + y.

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Use the given information to find each of the following.

sin y, given cos 2y = -1/3 , π/2 < y < π

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

-sin 35°

Use the given information to find cos(x - y).

sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

cot(-θ)/sec(-θ)

Use the given information to find the quadrant of x + y.

sin y = - 2/3, cos x = -1/5 , x in quadrant II, y in quadrant III

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

sin 35°

Use the given information to find tan(x + y).

cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III

Work each problem.

Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities. 

Use the given information to find each of the following.

sin 2x, given sin x = 0.6, π/2 < y < π

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos(-55°)

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

csc θ - sin θ

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

Use the given information to find sin(x + y).

sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant III

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

B, given cos 2B = 1/8 , 540° < 2B < 720°