Use the half-angle formulas to determine the exact value of the following.
$$\cos \left( - 22.5 ^ { \circ } \right)$$
A) $- \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 }$
B) $\frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }$
C) $- \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }$
D) $\frac { \sqrt { 3 - \sqrt { 2 } } } { 2 }$
E) $\frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }$

Question

Quizplus · Accepted Answer

We start with the half-angle formula for cosine:
$$\cos \frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}$$

We can rewrite the given angle as:
$$-22.5^{\circ}=-(45^{\circ}/2)=-(\pi/4)$$

Now, we look for the cosine of $-\pi/8$. To do that, we take $\theta=-\pi/4$ in the half-angle formula to get:
$$\cos\left(-\frac{\pi}{8}\right)=\sqrt{\frac{1+\cos\left(-\frac{\pi}{4}\right)}{2}}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}=\boxed{\frac{\sqrt{2+\sqrt{3}}}{2}}$$
The negative value of cosine is discarded because the given angle lies in the fourth quadrant, where cosine is positive.

Use the Half-Angle Formulas to Determine the Exact Value of the Following