Find the exact value by using a half-angle identity.
-$\sin 22.5 ^ { \circ }$
A) $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$
B) $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 2 } }$
C) $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }$
D) $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 2 } }$

Question

Quizplus · Accepted Answer

The half-angle formula for sine is $\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$. For $\sin(22.5^\circ)$, we use $\theta = 45^\circ$, so $\cos(45^\circ) = \frac{\sqrt{2}}{2}$. Plugging this into the formula gives $\sin(22.5^\circ) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \frac{1}{2}\sqrt{2 - \sqrt{2}}$.

Find the Exact Value by Using a Half-Angle Identity sin⁡22.5∘\sin 22.5 ^ { \circ }sin22.5∘

Find the Exact Value by Using a Half-Angle Identity $\sin 22.5 ^ { \circ }$