Question

write the expression as the sine or cosine of an angle. sin 3x cos x - cos 3x sin x ?( x) hint: sin(a ± b) = sin a cos b ± cos a sin b cos(a ± b) = cos a cos b ∓ sin a sin b

Explanation:

Step1: Recall the sine subtraction formula

The formula for \(\sin(A - B)=\sin A\cos B-\cos A\sin B\).

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

In the given expression \(\sin3x\cos x-\cos3x\sin x\), we can see that \(A = 3x\) and \(B=x\).

Step3: Apply the formula

Substituting \(A = 3x\) and \(B = x\) into the formula \(\sin(A - B)\), we get \(\sin(3x - x)\).

Step4: Simplify the expression

\(3x-x=2x\), so the expression is \(\sin(2x)\).

Answer:

\(\sin(2x)\)