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.cos² 105° - sin² 105°

Verified step by step guidance

1. Recognize that the expression \( \cos^2 105^\circ - \sin^2 105^\circ \) can be rewritten using the double angle identity for cosine: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \).
2. Identify \( \theta = 105^\circ \) in the expression, so the expression becomes \( \cos(2 \times 105^\circ) \).
3. Calculate the double angle: \( 2 \times 105^\circ = 210^\circ \).
4. Rewrite the expression as \( \cos(210^\circ) \).
5. Use the unit circle or known values to find the exact value of \( \cos(210^\circ) \).

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 cosine of a double angle can be expressed as cos(2θ) = cos²(θ) - sin²(θ). Understanding these formulas is essential for simplifying expressions involving angles that are multiples of a given angle.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is fundamental in trigonometry as it allows for the conversion between sine and cosine functions, facilitating the simplification of expressions and the evaluation of trigonometric values.

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°, 30°, 45°, 60°, and 90°. Knowing these values helps in quickly evaluating trigonometric expressions without the need for a calculator, which is particularly useful in problems involving double angles.