6. Trigonometric Identities and More Equations

Use the given information to find each of the following.

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

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

csc θ cos θ tan θ

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

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

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.)

If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

1/ tan² α + cot α tan α

Find sin θ.

cos θ = 5/6, θ in quadrant I

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

Verify each identity. cos θ sec θ/cot θ= tan θ

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

2y, given sec y = -5/3, sin y > 0

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°

In a $\mathrm{circle}$, if two $\mathrm{chords}$ are $\mathrm{congruent}$, what can be said about the $\mathrm{arcs}$ they subtend?