Question

Find the exact value by using a half-angle identity. (4 points) sin (7pi/8)

Answer 1

Answer:

$sin\frac{7\pi}{8}=\sqrt{\frac{\sqrt 2-1}{2\sqrt 2}}$

Step-by-step explanation:

We are given that

$sin\frac{7\pi}{8}$

We have to find the exact value by using a half angle identity.

Half angle identity:

$sin^2\theta=\frac{1-cos2\theta}{2}$

By using the formula

$sin\frac{7\pi}{8}=\sqrt{\frac{1-cos2(\frac{7\pi}{8})}{2}}$

$sin\frac{7\pi}{8}=\sqrt{\frac{1-cos\frac{7\pi}{4}}{2}}$

$sin\frac{7\pi}{8}=\sqrt{\frac{1-cos(2\pi-\frac{\pi}{4})}{2}}$

$sin\frac{7\pi}{8}=\sqrt{\frac{1-cos\frac{\pi}{4}}{2}}$

By using identity :$cos(2\pi-\theta)=cos\theta$

$sin\frac{7\pi}{8}=\sqrt{\frac{1-\frac{1}{\sqrt 2}}{2}}$

By using $cos\frac{\pi}{4}=\frac{1}{\sqrt 2}$

$sin\frac{7\pi}{8}=\sqrt{\frac{\sqrt 2-1}{2\sqrt 2}}$