Question

use the circle shown in the rectangular coordinate system to find two angles, in radians, between - 2π and 2π such that each angles terminal side passes through the origin and the point indicated on the circle. the two angles that determine the indicated point on the circle are. (simplify your answers. type exact answers in terms of π. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)

Explanation:

Step1: Recall angle - concept on unit - circle

Angles in the unit - circle are measured counter - clockwise from the positive x - axis. One full rotation is \(2\pi\) radians and a clockwise rotation gives negative angles.

Step2: Consider the position of the point on the circle

Without seeing the specific point on the circle, assume the general approach. If the point is in the first quadrant, a positive angle \(\theta\) between \(0\) and \(\frac{\pi}{2}\) can be found. To get a negative angle with the same terminal side, we subtract \(2\pi\) from the positive angle.
Let's assume the positive angle is \(\alpha\) and the negative angle is \(\beta\).
If the positive angle \(\alpha\) is given by the standard position of the terminal side of the angle in the counter - clockwise direction, then \(\beta=\alpha - 2\pi\) (as long as \(\beta\) is in the range \((- 2\pi,2\pi)\)).
For example, if the point corresponds to an angle of \(\frac{\pi}{4}\) in the positive direction (counter - clockwise), then the negative angle with the same terminal side is \(\frac{\pi}{4}-2\pi=\frac{\pi - 8\pi}{4}=-\frac{7\pi}{4}\)

Answer:

(The answer depends on the specific point on the circle. For illustration purposes, if the positive angle is \(\frac{\pi}{4}\), the two angles are \(-\frac{7\pi}{4},\frac{\pi}{4}\))