6. Trigonometric Identities and More Equations

Find one value of θ or x that satisfies each of the following.

sin θ = cos(2θ + 30°)

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin(45° + θ)

Find one value of θ or x that satisfies each of the following.

sec x = csc (2π/3)

Write each function as an expression involving functions of θ or x alone. See Example 2.

tan (π/4 + x)

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin (3π/4 - x)

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(90° - θ)

Write each function as an expression involving functions of θ or x alone. See Example 2.

tan(180° + θ)

Write each function as an expression involving functions of θ or x alone. See Example 2.

sin(π + x)

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

cos 3x

Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function of θ alone.

cos(270° + θ)

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

cos 4x

Find cos(s + t) and cos(s - t).

cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III

Find cos(s + t) and cos(s - t).

sin s = 2/3 and sin t = -1/3, s in quadrant II and t in quadrant IV

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

sin 60° cos 45° - cos 60° sin 45°

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos( π/2 + x) = -sin x