In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 𝝅 2 cos² ------ ﹣ 1 8

Key Concepts

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For example, the sine, cosine, and tangent double angle formulas are: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos²(θ) - sin²(θ), and tan(2θ) = 2tan(θ) / (1 - tan²(θ)). Understanding these formulas is essential for rewriting expressions involving double angles.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This fundamental identity allows us to express one trigonometric function in terms of another, which is particularly useful when simplifying expressions or solving equations. It can help in transforming expressions involving cos²(θ) into terms of sin²(θ) or vice versa.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent for commonly used angles, such as 0, π/6, π/4, π/3, and π/2. These values are often derived from the unit circle and can be used to evaluate trigonometric expressions without a calculator. Knowing these exact values is crucial for finding the exact value of expressions involving trigonometric functions.