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

Question

Find the exact value by using a half-angle identity.

sin(7
`\pi`/8)

Answer 1

`\bf sin\left(\cfrac{{{ \theta}}}{2}\right)=\pm \sqrt{\cfrac{1-cos({{ \theta}})}{2}}\\\\ -------------------------------\\\\ \cfrac{7\pi }{8}\cdot 2\implies \cfrac{7\pi }{4}\qquad \qquad sin\left((7\pi )/(8) \right)\implies sin\left(\cfrac{(7\pi )/(4)}{2} \right) \\\\\\ sin\left(\cfrac{(7\pi )/(4)}{2} \right)=\pm\sqrt{\cfrac{1-cos\left((7\pi )/(4) \right)}{2}}\implies sin\left(\cfrac{(7\pi )/(4)}{2} \right)=\pm\sqrt{\cfrac{1-(√(2))/(2)}{2}}`


`\bf sin\left(\cfrac{(7\pi )/(4)}{2} \right)=\pm\sqrt{\cfrac{(2-√(2))/(2)}{2}} \implies sin\left(\cfrac{(7\pi )/(4)}{2} \right)=\pm\sqrt{\cfrac{2-√(2)}{4}} \\\\\\ sin\left(\cfrac{(7\pi )/(4)}{2} \right)=\pm\cfrac{\sqrt{2-√(2)}}{√(4)}\implies sin\left(\cfrac{(7\pi )/(4)}{2} \right)=\pm\cfrac{\sqrt{2-√(2)}}{2}`