Question

Suppose cot(theta) = -8/7 and theta is in quadrant 4. find the value of csc(theta).

Answer 1

$\begin{gathered} \text{Given} \\ \cot \theta=-\frac{8}{7} \end{gathered}$

$\begin{gathered} \text{Recall that} \\ \cot \theta=\frac{\text{adjacent}}{\text{opposite}}, \\ \\ \text{given that }\theta\text{ is found in Quadrant 4,} \end{gathered}$

$\begin{gathered} \csc \theta\text{ is defined as} \\ \csc \theta=\frac{\text{hypotenuse}}{\text{opposite}} \end{gathered}$

Since we already know of the measure for the opposite side, we need to solve for the hypotenuse using Pythagorean theorem

$\begin{gathered} c^2=a^2+b^2 \\ c^2=(-7)^2+(8)^2 \\ c^2=49+64 \\ c^2=113 \\ \sqrt[]{c^2}=\sqrt[]{113} \\ c=\sqrt[]{113} \end{gathered}$

Now that we know the measure of the hypotenuse, substitute it to the definition for cosecant and we have

$\begin{gathered} \csc \theta=\frac{\text{hypotenuse}}{\text{opposite}} \\ \csc \theta=\frac{\sqrt[]{113}}{-7} \\ \\ \text{Therefore} \\ \csc \theta=-\frac{\sqrt[]{113}}{7} \end{gathered}$