Question

write the expression as the sine or cosine of an angle.
\\( \sin 3 x \cos x - \cos 3 x \sin x \\)
\\( \square ( \square x ) \\)
hint: \\( \sin ( a \pm b ) = \sin a \cos b \pm \cos a \sin b \\)
\\( \cos ( a \pm b ) = \cos a \cos b \mp \sin a \sin b \\)

Explanation:

Step1: Identify the formula

The given expression $\sin 3x\cos x-\cos 3x\sin x$ matches the formula $\sin(A - B)=\sin A\cos B-\cos A\sin B$. Here, $A = 3x$ and $B=x$.

Step2: Apply the formula

Substitute $A = 3x$ and $B = x$ into the formula $\sin(A - B)$. We get $\sin(3x - x)$.

Step3: Simplify the angle

Simplify $3x-x$. Using basic algebraic subtraction, $3x-x=(3 - 1)x=2x$.

Answer:

$\sin(2x)$