Question

Angle θ is in standard position and (−6,−3) is a point on the terminal side of θ. What is the exact value of csc θ in simplest form with a rational denominator?

Answer 1

Answer:

$\displaystyle \csc\theta=-\frac{\sqrt{5}}{2}$

Step-by-step explanation:

Since $\displaystyle \csc\theta=\frac{1}{\sin\theta}=\frac{1}{\frac{\text{Opposite}}{\text{Hypotenuse}}}=\frac{\text{Hypotenuse}}{\text{Opposite}}$, we need to find the hypotenuse given our corresponding opposite and adjacent lengths of -6 and -3 accounting for Quadrant III:

$(-6)^2+(-3)^2=c^2\\\\36+9=c^2\\\\45=c^2\\\\\sqrt{45}=c\\\\c=3\sqrt{5}$

Thus, $\displaystyle \csc\theta=\frac{\text{Hypotenuse}}{\text{Opposite}}=\frac{3\sqrt{5}}{-6}=-\frac{\sqrt{5}}{2}$