Write each expression as a product of trigonometric functions. See Example 8.
sin 102° - sin 95°

Verified step by step guidance

1. Recognize that the expression is a difference of sines: \(\sin 102^\circ - \sin 95^\circ\).
2. Recall the trigonometric identity for the difference of sines: \(\sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right)\).
3. Identify \(A = 102^\circ\) and \(B = 95^\circ\) and substitute these values into the identity.
4. Calculate the average of the angles: \(\frac{102^\circ + 95^\circ}{2}\) and the half difference: \(\frac{102^\circ - 95^\circ}{2}\), but do not simplify the numerical values yet.
5. Write the expression as a product using the identity: \(2 \cos \left( \frac{102^\circ + 95^\circ}{2} \right) \sin \left( \frac{102^\circ - 95^\circ}{2} \right)\).

Key Concepts

Sum-to-Product Identities

Sum-to-product identities transform sums or differences of sine and cosine functions into products. For example, the difference of sines can be expressed as 2 cos((A+B)/2) sin((A−B)/2). This simplifies expressions and is useful for solving or rewriting trigonometric problems.

Angle Measurement in Degrees

Trigonometric functions can take angles measured in degrees or radians. Understanding how to work with degrees, including converting or averaging angles, is essential when applying identities like sum-to-product, especially when angles are not standard special angles.

Basic Sine Function Properties

The sine function is periodic and odd, meaning sin(−θ) = −sin(θ). Recognizing these properties helps in manipulating expressions and applying identities correctly, ensuring accurate transformation of sums or differences of sine terms.