Solve the multiple-angle equation. $2 \sec 4 x = 4$
A) $\frac { \pi } { 12 } - \frac { n \pi } { 3 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$
B) $\frac { 3 \pi } { 4 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
C) $\frac { \pi } { 2 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
D) $\frac { 5 \pi } { 12 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
E) $\frac { \pi } { 12 } + \frac { n \pi } { 2 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$

Question

Solve the multiple-angle equation.​ $2 \sec 4 x = 4$ ​
A) $\frac { \pi } { 12 } - \frac { n \pi } { 3 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$
B)​ $\frac { 3 \pi } { 4 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
C)​ $\frac { \pi } { 2 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
D)​ $\frac { 5 \pi } { 12 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
E)​ $\frac { \pi } { 12 } + \frac { n \pi } { 2 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$

Quizplus · Accepted Answer

First, solve $2 \sec 4x = 4$ by dividing both sides by 2 to get $\sec 4x = 2$. This means $\cos 4x = \frac{1}{2}$. The solutions to $\cos \theta = \frac{1}{2}$ are $\theta = \frac{\pi}{3} + 2n\pi$ and $\theta = -\frac{\pi}{3} + 2n\pi$, but since we're dealing with $4x$, we divide these by 4 to get $x = \frac{\pi}{12} + \frac{n\pi}{2}$ and $x = -\frac{\pi}{12} + \frac{n\pi}{2}$. These solutions correspond to the angles where the cosine function equals $\frac{1}{2}$, taking into account the periodicity of the cosine function and the fact that it's positive in the first and fourth quadrants.

Solve the Multiple-Angle Equation 2sec⁡4x=42 \sec 4 x = 42sec4x=4 ​ A)

Solve the Multiple-Angle Equation $2 \sec 4 x = 4$
A)