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.)

$-225°$

Answer 1

To find the two positive angles and two negative angles that are co-terminal with $-225°$:

Using the concept of $\theta +\left(360°\right)n$, where $\theta$ is the reference angle and $n$ is an integers.

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

$\begin{array}{rcl}\text{Coterminal angle}& =& -225°+\left(360°\right)\times 1\\ & \Rightarrow & \text{Coterminal angle}=-225°+360°\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& 135°\end{array}$

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

$\begin{array}{rcl}\text{Coterminal angle}& =& -225°+\left(360°\right)\times 2\\ & \Rightarrow & \text{Coterminal angle}=-225°+720°\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& 495°\end{array}$

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

$\begin{array}{rcl}\text{Coterminal angle}& =& -225°+\left(360°\right)\times -1\\ & \Rightarrow & \text{Coterminal angle}=-225°-360°\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& -585°\end{array}$

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

$\begin{array}{rcl}\text{Coterminal angle}& =& -225°+\left(360°\right)\times -2\\ & \Rightarrow & \text{Coterminal angle}=-225°-720°\left[\text{Simplifying}\right]\\ \therefore \text{Coterminal angle}& =& -945°\end{array}$

Hence, the two positive angles are $\mathbf{135}\mathbf{°}\mathbf{,}\mathbf{}\mathbf{495}\mathbf{°}$ and the two negative angles are $\mathbf{-}\mathbf{585}\mathbf{°}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{945}\mathbf{°}$ that are co-terminal with $-225°$.