Question

The measure of an angle in standard position is given.

Find two positive angles and two negative angles that are co terminal with the given angle. (Enter your answers as a comma-separated list.)

$-\frac{\mathrm{\pi }}{4}$

Answer 1

To find the two positive angles and two negative angles that are co-terminal with $-\frac{\mathrm{\pi }}{4}$:

Using the concept of $\theta +2\mathrm{\pi }n$, where $\theta$ is the reference angle and $n$ is an integers.

Now, calculate first positive angles, substitute $n=1$ :

$\begin{array}{rcl}\text{Coterminal angle}& =& -\frac{\mathrm{\pi }}{4}+2\mathrm{\pi }\times 1\\ & \Rightarrow & \text{Coterminal angle}=-\frac{\mathrm{\pi }}{4}+2\mathrm{\pi }\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& \frac{7\mathrm{\pi }}{4}\end{array}$

Calculate second positive angles, substitute $n=2$ :

$\begin{array}{rcl}\text{Coterminal angle}& =& -\frac{\mathrm{\pi }}{4}+2\mathrm{\pi }\times 2\\ & \Rightarrow & \text{Coterminal angle}=-\frac{\mathrm{\pi }}{4}+4\mathrm{\pi }\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& \frac{15\mathrm{\pi }}{4}\end{array}$

Now, calculate first negative angles, substitute $n=-1$ :

$\begin{array}{rcl}\text{Coterminal angle}& =& -\frac{\mathrm{\pi }}{4}+2\mathrm{\pi }\times -1\\ & \Rightarrow & \text{Coterminal angle}=-\frac{\mathrm{\pi }}{4}-2\mathrm{\pi }\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& -\frac{9\mathrm{\pi }}{4}\end{array}$

Calculate second negative angles, substitute $n=-2$ :

$\begin{array}{rcl}\text{Coterminal angle}& =& -\frac{\mathrm{\pi }}{4}+2\mathrm{\pi }\times -2\\ & \Rightarrow & \text{Coterminal angle}=-\frac{\mathrm{\pi }}{4}-4\mathrm{\pi }\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& -\frac{17\mathrm{\pi }}{4}\end{array}$

Hence, the two positive angles are $\frac{\mathbf{7}\mathbf{\pi }}{\mathbf{4}}\mathbf{,}\mathbf{}\frac{\mathbf{15}\mathbf{\pi }}{\mathbf{4}}$ and the two negative angles are $-\frac{\mathbf{9}\mathbf{\pi }}{\mathbf{4}}\mathbf{,}\mathbf{}\mathbf{-}\frac{\mathbf{17}\mathbf{\pi }}{\mathbf{4}}$ that are co terminal with $-\frac{\mathrm{\pi }}{4}$.