Question

Given theta is an angle in Quadrant III such that $\sin \theta =-\frac{3}{5}$ How does one find the exact values of $\text{sec}\theta$ and $\text{cot}\theta$?

Answer 1

To find the exact values of $\text{sec}\theta$ and $\text{cot}\theta$.

Since $\sin \theta$ is negative, so the $\theta$ lies in the third quadrant.

Since sine function is ratio of $\frac{\text{Opposite}}{\text{Hypotenuse}}$, so the opposite is $3$ and the hypotenuse is $5$.

Using the Pythagorean Theorem, find the value of Adjacent:

$\begin{array}{rcl}{\left(\text{Hypotenuse}\right)}^2& =& {\left(\text{Opposite}\right)}^2+{\left(\text{Adjacent}\right)}^2\left[\text{Use Pythagorean Theorem}\right]\\ & \Rightarrow & {\left(5\right)}^2={\left(3\right)}^2+{\left(\text{Adjacent}\right)}^2\left[\text{Substitute all values}\right]\\ & \Rightarrow & 25=9+{\left(\text{Adjacent}\right)}^2\left[\text{Simplifying}\right]\\ & \Rightarrow & {\left(\text{Adjacent}\right)}^2=25-9\\ & \Rightarrow & {\left(\text{Adjacent}\right)}^2=16\\ & \Rightarrow & \text{Adjacent}=\sqrt{16}\left[\text{Take square root}\right]\\ \therefore \text{Adjacent}& =& 4\end{array}$

Now, find the value of $\text{sec}\theta$:

$\begin{array}{rcl}\text{sec}\theta & =& \frac{\text{Hypotenuse}}{\text{Adjacent}}\\ \therefore \text{sec}\theta & =& \frac{5}{4}\end{array}$

Since $\theta$ lies in the third quadrant, so the secant function is negative, $\text{sec}\theta =-\frac{5}{4}$.

Now, find the value of $\text{cot}\theta$:

$\begin{array}{rcl}\text{cot}\theta & =& \frac{\text{Adjacent}}{\text{Opposite}}\\ \therefore \text{cot}\theta & =& \frac{4}{3}\end{array}$

Since $\theta$ lies in the third quadrant, so the cotangent function is positive, $\text{cot}\theta =\frac{4}{3}$.

Hence, the value of $\text{sec}\theta$ is $\mathbf{-}\frac{\mathbf{5}}{\mathbf{4}}$ and $\text{cot}\theta$ is $\frac{\mathbf{4}}{\mathbf{3}}$.