use a sum - to - product identity to rewrite the expression.\n$sin 3alpha+sin 6alpha$\n$sin 3alpha+sin…

use a sum - to - product identity to rewrite the expression.\n$sin 3alpha+sin 6alpha$\n$sin 3alpha+sin 6alpha=square$\n(use integers or fractions for any numbers in the expression.)

use a sum - to - product identity to rewrite the expression.\n$sin 3alpha+sin 6alpha$\n$sin 3alpha+sin 6alpha=square$\n(use integers or fractions for any numbers in the expression.)

Answer

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 = 3\alpha) and (B=6\alpha).

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

(\frac{A + B}{2}=\frac{3\alpha+6\alpha}{2}=\frac{9\alpha}{2}) and (\frac{A - B}{2}=\frac{3\alpha - 6\alpha}{2}=-\frac{3\alpha}{2}). Since (\cos(-x)=\cos x), we can use (\cos\frac{3\alpha}{2}) instead of (\cos(-\frac{3\alpha}{2})).

Step4: Substitute into the formula

(\sin3\alpha+\sin6\alpha = 2\sin\frac{9\alpha}{2}\cos\frac{3\alpha}{2})

Answer:

(2\sin\frac{9\alpha}{2}\cos\frac{3\alpha}{2})