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

Question

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

Quizplus · Accepted Answer

Using the half-angle formula for cosine, we have:$$\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$$Substituting $\cos x = \frac{1}{4}$, we get:$$\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \frac{1}{4}}{2}} = \pm \sqrt{\frac{5}{8}} = \pm \frac{\sqrt{5}}{4}$$Since $x$ terminates in $0 < x < \frac{\pi}{2}$, we have $\cos x > 0$, so the positive solution is the correct one:$$\cos \frac{x}{2} = \frac{\sqrt{5}}{4}$$

Determine All Solutions of the Equation in Radians cos⁡x2\cos \frac { x } { 2 }cos2x​

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