6. Trigonometric Identities and More Equations

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.

sin 158.2°/(1 + cos 158.2°)

Simplify each expression.

±√[(1 + cos 18x)/2]

Simplify each expression.

±√[(1 + cos 20α)/2]

Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

Simplify each expression.

±√[(1 - cos 5A)/(1 + cos 5A)]

Simplify each expression.

± √[(1 + cos (x/4))/2]

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.

tan (θ/2) = csc θ - cot θ

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°