Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sin \theta = \frac { 1 } { 4 } , \quad 0

Question

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sin \theta = \frac { 1 } { 4 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$ 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 $\sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}}$. First, find $\cos \theta$ using $\sin^2 \theta + \cos^2 \theta = 1$, which gives $\cos \theta = \sqrt{1 - \left(\frac{1}{4}\right)^2} = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4}$. Then, $\sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{\sqrt{15}}{4}}{2}} = \sqrt{\frac{8 - 2\sqrt{15}}{16}} = \frac{\sqrt{8 - 2\sqrt{15}}}{4}$.

Use the Information Given About the Angle θ, 0 ≤θ sin⁡θ=14,0<θ<π2\sin \theta = \frac { 1 } { 4 } , \quad 0 < \theta < \frac { \pi } { 2 } \quadsinθ=41​,0<θ<2π​

Use the Information Given About the Angle θ, 0 ≤θ $\sin \theta = \frac { 1 } { 4 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$