Find the exact value by using a half-angle identity.
-$\cos 75 ^ { \circ }$
A) $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$
B) $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$
C) $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$
D) $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$

Question

Quizplus · Accepted Answer

The half-angle formula for cosine is $\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$. For $\cos(75^\circ)$, we use $\theta = 150^\circ$, so $\cos(150^\circ) = -\frac{\sqrt{3}}{2}$. Plugging into the formula gives $\cos(75^\circ) = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \frac{1}{2}\sqrt{2 - \sqrt{3}}$.

Find the Exact Value by Using a Half-Angle Identity cos⁡75∘\cos 75 ^ { \circ }cos75∘

Find the Exact Value by Using a Half-Angle Identity $\cos 75 ^ { \circ }$