6. Trigonometric Identities and More Equations

Use identities to fill in each blank with the appropriate trigonometric function name.

sin 2π/3 = _____ (- π/6)

Find the exact value of each expression. See Example 1.

sin 40° cos 50° + cos 40° sin 50°

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°

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

cos(θ - 270°)

Find the exact value of each expression.

sin (13π/12)

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin(60° - 45°)

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

cos 2x = (cot² x - 1)/(cot² x + 1)

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. sin 75°

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 sin(s + t). See Example 3.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

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

cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV

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

tan(180° + θ)

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

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

tan α = ﹣3/4, α lies in quadrant II, and cos β = 1/3, β lies in quadrant I.