Determine all solutions of the equation in radians.
-Find $\cos \frac { \theta } { 2 }$, given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0

Question

Determine all solutions of the equation in radians.
-Find $\cos \frac { \theta } { 2 }$, given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$.
A) $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$
B) $\frac { \sqrt { 10 } } { 4 }$
C) $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$
D) $\frac { \sqrt { 6 } } { 4 }$

Quizplus · Accepted Answer

Given $\sin \theta = \frac{1}{4}$ and $\theta$ is in the first quadrant, we use the half-angle formula for cosine: $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$. Since $\theta$ is in the first quadrant, $\cos \frac{\theta}{2} > 0$. We find $\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{1}{4}\right)^2} = \sqrt{\frac{15}{16}}$, leading to $\cos \frac{\theta}{2} = \sqrt{\frac{1 + \sqrt{\frac{15}{16}}}{2}} = \frac{\sqrt{8 + 2\sqrt{15}}}{4}$.

Determine All Solutions of the Equation in Radians cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​

Determine All Solutions of the Equation in Radians $\cos \frac { \theta } { 2 }$