Find the exact value. $\operatorname { COS } \frac { 5 \pi } { 12 }$
A) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
B) $\frac { 5 \pi } { 12 }$
C) $\frac { \sqrt { 6 } } { 4 }$
D) $\frac { \sqrt { 3 } } { 2 }$
E) none

Question

Quizplus · Accepted Answer

We can use the half-angle formula for cosine:

$$\cos\frac{5\pi}{12}=\sqrt{\frac{1+\cos\frac{5\pi}{6}}{2}}=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$$

Simplifying the numerator gives:

$$=\sqrt{\frac{2+\sqrt{3}}{4}}=\sqrt{\frac{1}{2}+\frac{\sqrt{3}}{4}}$$

Now we rationalize the denominator by multiplying numerator and denominator by $\sqrt{2}$:

$$\sqrt{\frac{1}{2}+\frac{\sqrt{3}}{4}}=\sqrt{\frac{2}{4}+\frac{\sqrt{3}}{4}}=\sqrt{\frac{2+\sqrt{3}}{4}}=\frac{\sqrt{2+\sqrt{3}}}{2}$$

Finally, we simplify using the difference of squares:

$$\frac{\sqrt{2+\sqrt{3}}}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}$$

Therefore, the answer is $\boxed{\textbf{(A)}}$.

Find the Exact Value COS⁡5π12\operatorname { COS } \frac { 5 \pi } { 12 }COS125π​

Find the Exact Value $\operatorname { COS } \frac { 5 \pi } { 12 }$