Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \frac { x } { 2 }$, given that $\sin x = \frac { 1 } { 4 }$ and $x$ terminates in $0

Question

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \frac { x } { 2 }$, given that $\sin x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.
A) $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$
B) $\frac { \sqrt { 6 } } { 4 }$
C) $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$
D) $\frac { \sqrt { 10 } } { 4 }$

Quizplus · Accepted Answer

Using the half-angle formula for sine, we have:$$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$$Since $x$ terminates in $0 < x < \frac{\pi}{2}$, $\cos x > 0$. Therefore, we have:$$\sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}}$$Substituting $\sin x = \frac{1}{4}$, we get:$$\sin \frac{x}{2} = \sqrt{\frac{1 - \sqrt{1 - \sin^2 x}}{2}}$$$$\sin \frac{x}{2} = \sqrt{\frac{1 - \sqrt{1 - \left(\frac{1}{4}\right)^2}}{2}}$$$$\sin \frac{x}{2} = \sqrt{\frac{1 - \sqrt{\frac{15}{16}}}{2}}$$$$\sin \frac{x}{2} = \sqrt{\frac{1 - \frac{\sqrt{15}}{4}}{2}}$$$$\sin \frac{x}{2} = \sqrt{\frac{4 - \sqrt{15}}{8}}$$$$\sin \frac{x}{2} = \frac{\sqrt{8 - 2\sqrt{15}}}{4}$$Therefore, the answer is A.

Use the Fundamental Identities to Find the Value of the Trigonometric