Find the exact value by using a half-angle identity.
-$\sin \frac { 7 \pi } { 8 }$
A) $- \frac { 1 } { 2 } \sqrt { 1 - \sqrt { 2 } }$
B) $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$
C) $\frac { 1 } { 2 } \sqrt { 1 - \sqrt { 2 } }$
D) $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$

Question

Quizplus · Accepted Answer

The half-angle identity for sine is $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$. For $\theta = \frac{7\pi}{4}$, we find $\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}$. Substituting into the half-angle formula gives $\sin \frac{7\pi}{8} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \frac{1}{2}\sqrt{2 - \sqrt{2}}$.

Find the Exact Value by Using a Half-Angle Identity sin⁡7π8\sin \frac { 7 \pi } { 8 }sin87π​

Find the Exact Value by Using a Half-Angle Identity $\sin \frac { 7 \pi } { 8 }$