Question

use the fundamental identities to find $csc\theta$ if $sin\theta =-\frac{2}{3}$ and $\theta$ is in quadrant iv.
a. $\frac{5}{4}$
b. $\frac{3sqrt{7}}{7}$
c. $-\frac{3}{2}$
d. $-\frac{sqrt{7}}{9}$

Explanation:

Step1: Recall the reciprocal identity

The reciprocal identity for cosecant and sine is $\csc\theta=\frac{1}{\sin\theta}$.

Step2: Substitute the given value of $\sin\theta$

Given $\sin\theta =-\frac{2}{3}$, then $\csc\theta=\frac{1}{-\frac{2}{3}}$.

Step3: Simplify the expression

$\frac{1}{-\frac{2}{3}}=-\frac{3}{2}$.

Answer:

C. $-\frac{3}{2}$