Use the given information to find the exact value of the trigonometric function.
-$\cos \theta = \frac { 1 } { 4 } , \csc \theta 0 \quad$ Find $\sin \frac { \theta } { 2 }$.
A) $\frac { \sqrt { 6 } } { 4 }$
B) $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$
C) $\frac { \sqrt { 10 } } { 4 }$
D) $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$

Question

Use the given information to find the exact value of the trigonometric function.
-$\cos \theta = \frac { 1 } { 4 } , \csc \theta > 0 \quad$ Find $\sin \frac { \theta } { 2 }$.
A) $\frac { \sqrt { 6 } } { 4 }$
B) $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$
C) $\frac { \sqrt { 10 } } { 4 }$
D) $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$

Quizplus · Accepted Answer

The half-angle formula for sine is $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$. Given $\cos \theta = \frac{1}{4}$ and $\csc \theta > 0$ implies $\theta$ is in the first or second quadrant, where sine is positive. Thus, $\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{4}}{2}} = \sqrt{\frac{3}{8}} = \frac{\sqrt{6}}{4}$.

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