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

Verified step by step guidance

1. Recognize that the expression involves the product of sine and cosine of the same angle: \(\frac{1}{8} \sin 29.5^\circ \cos 29.5^\circ\).
2. Recall the double-angle identity for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\). This allows us to rewrite the product \(\sin \theta \cos \theta\) as \(\frac{1}{2} \sin 2\theta\).
3. Apply this identity to the expression: replace \(\sin 29.5^\circ \cos 29.5^\circ\) with \(\frac{1}{2} \sin (2 \times 29.5^\circ)\), which simplifies to \(\frac{1}{2} \sin 59^\circ\).
4. Substitute back into the original expression: \(\frac{1}{8} \times \frac{1}{2} \sin 59^\circ\).
5. Combine the constants: multiply \(\frac{1}{8}\) and \(\frac{1}{2}\) to get \(\frac{1}{16}\), so the simplified expression is \(\frac{1}{16} \sin 59^\circ\).

Key Concepts

Basic Trigonometric Functions

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. Understanding their values and properties is essential for simplifying expressions involving these functions.

Product-to-Sum Identities

Product-to-sum identities transform products of sine and cosine into sums or differences of trigonometric functions. For example, sin A cos B can be rewritten as ½ [sin(A + B) + sin(A - B)], which simplifies calculations and expressions.

Angle Measurement and Degree Conversion

Trigonometric functions often require angle inputs in degrees or radians. Knowing how to interpret and convert these measurements ensures accurate evaluation and simplification of expressions involving specific angle values.