Find the exact value by using a half-angle identity.
-$\cos 22.5 ^ { \circ }$
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 cosine is $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$. For $\cos 22.5^\circ$, we use $\theta = 45^\circ$, so $\cos 45^\circ = \frac{\sqrt{2}}{2}$. Plugging this into the formula gives $\cos 22.5^\circ = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \frac{1}{2}\sqrt{2 + \sqrt{2}}$.

Find the Exact Value by Using a Half-Angle Identity cos⁡22.5∘\cos 22.5 ^ { \circ }cos22.5∘

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