Question

express the given sum or difference as a product of sines and/or cosines. sin 3x + sin 7x

Explanation:

Step1: Recall the sum - to - product formula

The sum - to - product formula for \(\sin A+\sin B\) is \(2\sin\frac{A + B}{2}\cos\frac{A - B}{2}\).

Step2: Identify \(A\) and \(B\)

Here \(A = 3x\) and \(B=7x\).

Step3: Calculate \(\frac{A + B}{2}\) and \(\frac{A - B}{2}\)

\(\frac{A + B}{2}=\frac{3x+7x}{2}=\frac{10x}{2} = 5x\)
\(\frac{A - B}{2}=\frac{3x - 7x}{2}=\frac{-4x}{2}=-2x\)
Since \(\cos(-\alpha)=\cos\alpha\), \(\cos(-2x)=\cos(2x)\)

Step4: Apply the formula

\(\sin3x+\sin7x=2\sin\frac{3x + 7x}{2}\cos\frac{3x - 7x}{2}=2\sin5x\cos2x\)

Answer:

\(2\sin5x\cos2x\)