use the half - angle formulas to find the exact value of the expression sin(-3π/8). sin(-3π/8)=-√(2 + √2)/2…

use the half - angle formulas to find the exact value of the expression sin(-3π/8). sin(-3π/8)=-√(2 + √2)/2 (type an exact answer, using radicals as needed. use integers or fractions for any numbers in the exp)

use the half - angle formulas to find the exact value of the expression sin(-3π/8). sin(-3π/8)=-√(2 + √2)/2 (type an exact answer, using radicals as needed. use integers or fractions for any numbers in the exp)

Answer

Explanation:

Step1: Recall half - angle formula for sine

$\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$. Let $\alpha=-\frac{3\pi}{4}$, then $\frac{\alpha}{2}=-\frac{3\pi}{8}$.

Step2: Find the value of $\cos\alpha$

We know that $\cos(-\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$.

Step3: Substitute into half - angle formula

$\sin(-\frac{3\pi}{8})=-\sqrt{\frac{1-\cos(-\frac{3\pi}{4})}{2}}=-\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}=-\frac{\sqrt{2+\sqrt{2}}}{2}$. The negative sign is because $-\frac{3\pi}{8}$ is in the fourth - quadrant where sine is negative.

Answer:

$-\frac{\sqrt{2+\sqrt{2}}}{2}$