Find all solutions of the following equation in the interval [0, 2$\pi $).
$3 \sin x + \csc x = 0$
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 $). 
$3 \sin x + \csc x = 0$  
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 equation $3 \sin x + \csc x = 0$ simplifies to $3 \sin x + \frac{1}{\sin x} = 0$. Multiplying both sides by $\sin x$ to clear the fraction gives $3 \sin^2 x + 1 = 0$, which is impossible since $\sin^2 x$ is always non-negative and adding 1 makes it strictly positive. Therefore, there are no solutions.

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