Question

express the given sum or difference as a product of sines and/or cosines.
\\( \sin 9 \theta - \sin 3 \theta \\)
\\( \sin 9 \theta - \sin 3 \theta = \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the express

Explanation:

Step1: Recall the sine subtraction formula

The formula for \(\sin A-\sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}\). Here \(A = 9\theta\) and \(B=3\theta\).

Step2: Substitute \(A\) and \(B\) into the formula

First, calculate \(\frac{A + B}{2}=\frac{9\theta+3\theta}{2}=\frac{12\theta}{2} = 6\theta\) and \(\frac{A - B}{2}=\frac{9\theta - 3\theta}{2}=\frac{6\theta}{2}=3\theta\).
Then, substitute into the formula: \(\sin9\theta-\sin3\theta=2\cos6\theta\sin3\theta\).

Answer:

\(2\cos6\theta\sin3\theta\)