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 the equation $2 \sec 4x = 4$. This simplifies to $\sec 4x = 2$, which means $\cos 4x = \frac{1}{2}$. The solutions to $\cos \theta = \frac{1}{2}$ are $\theta = \pm \frac{\pi}{3} + 2n\pi$, where $n$ is an integer. However, since we are solving for $4x$, we divide these solutions by 4, giving $x = \pm \frac{\pi}{12} + \frac{n\pi}{2}$. This matches the solutions given in option E, considering the periodicity of the cosine function and its positive value at these specific angles.

Solve the Multiple-Angle Equation 2sec⁡4x=42 \sec 4 x = 42sec4x=4 A) π12−nπ3,5π12+nπ2\frac { \pi } { 12 } - \frac { n \pi } { 3 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }12π​−3nπ​,125π​+2nπ​

Solve the Multiple-Angle Equation $2 \sec 4 x = 4$ A) $\frac { \pi } { 12 } - \frac { n \pi } { 3 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$