Write each expression as a product of trigonometric functions. See Example 8.
sin 9x - sin 3x

Verified step by step guidance

1. Recognize that the expression \( \sin 9x - \sin 3x \) is a difference of sines, which can be rewritten using the trigonometric identity for the difference of sines: \[ \sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \]
2. Identify \( A = 9x \) and \( B = 3x \) in the given expression.
3. Calculate the average of \( A \) and \( B \): \[ \frac{9x + 3x}{2} = \frac{12x}{2} = 6x \]
4. Calculate half the difference of \( A \) and \( B \): \[ \frac{9x - 3x}{2} = \frac{6x}{2} = 3x \]
5. Substitute these values back into the identity to express the original expression as a product: \[ \sin 9x - \sin 3x = 2 \cos 6x \sin 3x \]

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). These identities simplify expressions and are useful in integration and solving equations.

Trigonometric Function Properties

Understanding the basic properties of sine and cosine functions, such as periodicity and symmetry, helps in manipulating and simplifying expressions. Recognizing how angles combine and relate is essential for applying identities correctly.

Angle Manipulation and Substitution

Breaking down complex angles into sums or differences allows the use of identities effectively. For instance, expressing 9x and 3x in terms of their average and difference facilitates applying sum-to-product formulas to rewrite the expression as a product.