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)