Question

draw the terminal side of this angle. make sure to draw you ray to the end of the graph. 585°

Explanation:

Step1: Find the coterminal angle

To find the coterminal angle of \(585^{\circ}\), subtract \(360^{\circ}\) (one full - rotation).
\(585^{\circ}-360^{\circ}=225^{\circ}\)

Step2: Analyze the quadrant of the coterminal angle

Since \(180^{\circ}<225^{\circ}<270^{\circ}\), the angle \(225^{\circ}\) (and thus \(585^{\circ}\)) has its terminal side in the third quadrant.
The terminal side of a \(225^{\circ}\) (or \(585^{\circ}\)) angle makes an angle of \(225 - 180=45^{\circ}\) with the negative \(x\) - axis.

Answer:

Draw a ray in the third quadrant such that it makes a \(45^{\circ}\) angle with the negative \(x\) - axis (extending to the end of the graph).