Answer:
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{\sqrt 2-1}{2\sqrt 2}}[/tex]
Step-by-step explanation:
We are given that
[tex] sin\frac{7\pi}{8}[/tex]
We have to find the exact value by using a half angle identity.
Half angle identity:
[tex] sin^2\theta=\frac{1-cos2\theta}{2}[/tex]
By using the formula
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{1-cos2(\frac{7\pi}{8})}{2}}[/tex]
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{1-cos\frac{7\pi}{4}}{2}}[/tex]
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{1-cos(2\pi-\frac{\pi}{4})}{2}}[/tex]
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{1-cos\frac{\pi}{4}}{2}}[/tex]
By using identity :[tex]cos(2\pi-\theta)=cos\theta[/tex]
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{1-\frac{1}{\sqrt 2}}{2}}[/tex]
By using [tex] cos\frac{\pi}{4}=\frac{1}{\sqrt 2}[/tex]
[tex]sin\frac{7\pi}{8}=\sqrt{\frac{\sqrt 2-1}{2\sqrt 2}}[/tex]
