Question

express the sum or difference as a product. if possible, find the exact value of this product.

$$ sin left( \frac { 7 pi } { 12 } ight) - sin left( \frac { pi } { 12 } ight) $$

express the difference as a product.

$$ sin left( \frac { 7 pi } { 12 } ight) - sin left( \frac { pi } { 12 } ight) = - \frac { sqrt { 2 } } { 2 } $$

Explanation:

Step1: Apply the sine subtraction formula

The formula for \(\sin A-\sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}\). Here \(A=\frac{7\pi}{12}\) and \(B=\frac{\pi}{12}\).

$$
LATEXBLOCK0
$$

Step2: Simplify the trigonometric values

We know that \(\cos\frac{\pi}{3}=\frac{1}{2}\) and \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\).

$$
LATEXBLOCK1
$$

Answer:

\(\sin\frac{7\pi}{12}-\sin\frac{\pi}{12}=\frac{\sqrt{2}}{2}\)