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

`\displaystyle \csc\theta=-(√(5))/(2)`

Explanation:

Since
`\displaystyle \csc\theta=(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\\\\√(45)=c\\\\c=3√(5)`

Thus,
`\displaystyle \csc\theta=\frac{\text{Hypotenuse}}{\text{Opposite}}=(3√(5))/(-6)=-(√(5))/(2)`