Question

the point $p(x,y)$ is on the terminal ray of angle $\theta$. if $\theta$ is between $pi$ radians and $\frac{3pi}{2}$ radians and $csc\theta = -\frac{5}{2}$, what are the coordinates of $p(x,y)$?
$p(-sqrt{21}, - 2)$
$p(sqrt{21}, - 2)$
$p(-2,sqrt{21})$
$p(-2,-sqrt{21})$

Explanation:

Step1: Recall the definition of cosecant

We know that $\csc\theta=\frac{r}{y}$, and given $\csc\theta =-\frac{5}{2}$, so $r = 5$ and $y=- 2$ (since $\csc\theta=\frac{r}{y}$ and $r>0$).

Step2: Use the Pythagorean identity $x^{2}+y^{2}=r^{2}$

Substitute $r = 5$ and $y=-2$ into $x^{2}+y^{2}=r^{2}$, we get $x^{2}+(-2)^{2}=5^{2}$, which simplifies to $x^{2}+4 = 25$, then $x^{2}=21$, so $x=\pm\sqrt{21}$.

Step3: Determine the sign of $x$

Since $\theta$ is between $\pi$ and $\frac{3\pi}{2}$ radians, the angle is in the third - quadrant where $x<0$ and $y < 0$. So $x=-\sqrt{21}$.

Answer:

$P(-\sqrt{21},-2)$