Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$2 \sqrt { 3 } \sin 4 x = 3$
A) $\{ 0 \}$
B) $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$
C) $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$
D) $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$

Question

Quizplus · Accepted Answer

First, solve for $\sin 4x = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}$. The solutions for $\sin \theta = \frac{\sqrt{3}}{2}$ within one cycle [0, $2\pi$] are $\theta = \frac{\pi}{3}$ and $\theta = \frac{2\pi}{3}$. Since we're solving for $4x$, we consider $4x = \frac{\pi}{3} + 2k\pi$ and $4x = \frac{2\pi}{3} + 2k\pi$ for $k = 0, 1, 2, ...$. Solving for $x$ gives the eight solutions listed in option D within the interval [0, $2\pi$).

Solve the Equation for Solutions in the Interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π)

Solve the Equation for Solutions in the Interval [0, 2 $[ 0,2 \pi )$