Simplify each expression. See Example 4.
⅛ sin 29.5° cos 29.5°

Verified step by step guidance

1. Recognize that the expression \( \frac{1}{8} \sin 29.5^\circ \cos 29.5^\circ \) can be simplified using a trigonometric identity.
2. Recall the double angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \).
3. Set \( \theta = 29.5^\circ \), so \( 2\theta = 59^\circ \).
4. Rewrite \( \sin 29.5^\circ \cos 29.5^\circ \) as \( \frac{1}{2} \sin 59^\circ \) using the identity.
5. Substitute back into the original expression: \( \frac{1}{8} \times \frac{1}{2} \sin 59^\circ \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. A key identity relevant here is the double angle identity for sine, which states that sin(2θ) = 2sin(θ)cos(θ). This identity can simplify expressions involving products of sine and cosine functions.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. For an angle θ, sin(θ) is the ratio of the opposite side to the hypotenuse, while cos(θ) is the ratio of the adjacent side to the hypotenuse. Understanding these functions is essential for simplifying trigonometric expressions.

Angle Measurement

Angle measurement in degrees is a way to quantify the size of an angle. In this context, 29.5° is a specific angle that can be used in trigonometric calculations. It's important to be comfortable converting between degrees and radians, as well as understanding how to evaluate trigonometric functions at specific angles to simplify expressions effectively.