Find all solutions of the following equation in the interval $[ 0,2 \pi )$ $4 \sin x + \csc x = 0$
A) $\frac { 5 \pi } { 2 } + 2 \pi , \frac { 5 \pi } { 2 } + 2 \pi$
B) No solution
C) $\frac { \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
D) $\frac { 3 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
E) $\frac { 7 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$

Question

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ $4 \sin x + \csc x = 0$ 
A) $\frac { 5 \pi } { 2 } + 2 \pi , \frac { 5 \pi } { 2 } + 2 \pi$
B) No solution
C) $\frac { \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
D) $\frac { 3 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
E) $\frac { 7 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$

Quizplus · Accepted Answer

The given equation is $4 \sin x + \csc x = 0$. Rewriting $\csc x$ as $1/\sin x$, we get $4 \sin x + 1/\sin x = 0$. Multiplying through by $\sin x$ to clear the fraction gives $4 \sin^2 x + 1 = 0$, which is impossible since $\sin^2 x$ is always non-negative, and thus the left side is always greater than or equal to 1. Therefore, there are no solutions.

Find All Solutions of the Following Equation in the Interval [0,2π)[ 0,2 \pi )[0,2π)

Find All Solutions of the Following Equation in the Interval $[ 0,2 \pi )$