6. Trigonometric Identities and More Equations

Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12

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

cos 2x = 1 - 2 sin² x

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

a. cos (α + β)

tan α = 3/4, 𝝅 < α < 3𝝅/2, and cos β = 1/4, 3𝝅/2 < β < 2𝝅

Find the exact value of the expression.

$\(\tan\)105\(\degree\)$

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

tan (π/4 + x)

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

tan θ = cot(45° + 2θ)

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

Find the exact value of the expression.

$\(\cos\)105\(\degree\)$

Write each function value in terms of the cofunction of a complementary angle.

tan 174° 03'

Find the exact value of each expression.

sin 255°

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.

$\cos \frac{5\pi }{12}\cos \frac{\pi }{12}+\sin \frac{5\pi }{12}\sin \frac{\pi }{12}$

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

29. sin(5𝝅/12) cos(𝝅/4) - cos(5𝝅/12) sin(𝝅/4)

Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of the angle will be positive.) Use a calculator, and round to the nearest tenth of a degree.

5x - 2y + 4 = 0, 3x + 5y = 6