Use an identity to write the expression as a single trigonometric function or as a single number.
-Find $\sin \frac { \theta } { 2 }$, given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0

Question

Use an identity to write the expression as a single trigonometric function or as a single number.
-Find $\sin \frac { \theta } { 2 }$, given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$.
A) $\frac { \sqrt { 6 } } { 4 }$
B) $\frac { \sqrt { 10 } } { 4 }$
C) $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$
D) $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$

Quizplus · Accepted Answer

Using the half-angle identity for sine, we have:$$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$Since $0 < \theta < 90^\circ$, we have $\cos \theta > 0$, so we take the positive square root. Substituting $\cos \theta = \frac{1}{4}$, we get:$$\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{1}{4}}{2}} = \sqrt{\frac{3}{8}} = \frac{\sqrt{3}}{2\sqrt{2}} = \frac{\sqrt{6}}{4}$$

Use an Identity to Write the Expression as a Single sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​

Use an Identity to Write the Expression as a Single $\sin \frac { \theta } { 2 }$