Question

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

Answer 1

`sin(7\pi)/(8)=\sqrt{(\sqrt 2-1)/(2\sqrt 2)}`

Explanation:

We are given that

`sin(7\pi)/(8)`

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

Half angle identity:

`sin^2\theta=(1-cos2\theta)/(2)`

By using the formula

`sin(7\pi)/(8)=\sqrt{(1-cos2((7\pi)/(8)))/(2)}`

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

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

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

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

`sin(7\pi)/(8)=\sqrt{(1-(1)/(\sqrt 2))/(2)}`

By using
`cos(\pi)/(4)=(1)/(\sqrt 2)`

`sin(7\pi)/(8)=\sqrt{(\sqrt 2-1)/(2\sqrt 2)}`