Use the given information to find the exact value of the trigonometric function.
-$\sec \theta = 4 , \theta$ lies in quadrant I $\quad$ Find $\cos \frac { \theta } { 2 }$
A) $\frac { \sqrt { 10 } } { 4 }$
B) $\frac { \sqrt { 6 } } { 4 }$
C) $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$
D) $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$

Question

Quizplus · Accepted Answer

The half-angle formula for cosine is $\cos \frac { \theta } { 2 } = \pm \sqrt { \frac { 1 + \cos \theta } { 2 } }$. Since $\theta$ lies in quadrant I, $\cos \theta$ is positive. Therefore, $\cos \frac { \theta } { 2 } = \sqrt { \frac { 1 + \cos \theta } { 2 } }$. Substituting $\sec \theta = 4$ into this equation, we get $\cos \frac { \theta } { 2 } = \sqrt { \frac { 1 + \frac { 1 } { 4 } } { 2 } } = \sqrt { \frac { 5 } { 8 } } = \frac { \sqrt { 5 } } { 2 \sqrt { 2 } } = \frac { \sqrt { 10 } } { 4 }$.

Use the Given Information to Find the Exact Value of the Trigonometric