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 - rotation concept

Angles in standard position are measured counter - clockwise from the positive x - axis. One full rotation is \(2\pi\) radians.

Step2: Consider positive and negative angles

Positive angles are measured counter - clockwise and negative angles are measured clockwise. Let's assume the point on the circle is at a certain position. If we consider the shortest positive angle \(\theta_1\) and the shortest negative angle \(\theta_2\) that have their terminal sides passing through the point.
For example, if the point corresponds to a quarter - turn counter - clockwise from the positive x - axis, the positive angle \(\theta_1=\frac{\pi}{2}\) and the negative angle \(\theta_2 =-\frac{3\pi}{2}\) (since \(2\pi-\frac{\pi}{2}=\frac{3\pi}{2}\) in the clockwise direction).

However, since the point on the circle is not given in the problem description, in general, if the reference angle is \(\alpha\) (where \(0\leq\alpha < 2\pi\)), the two angles \(\theta_1=\alpha\) and \(\theta_2=\alpha - 2\pi\) (if \(\alpha>0\)) or \(\theta_1=\alpha + 2\pi\) (if \(\alpha<0\)) will satisfy the condition of being between \(- 2\pi\) and \(2\pi\).

Let's assume the point is at an angle \(\frac{\pi}{4}\) counter - clockwise from the positive x - axis.
The positive angle \(\theta_1=\frac{\pi}{4}\) and the negative angle \(\theta_2=\frac{\pi}{4}-2\pi=\frac{\pi - 8\pi}{4}=-\frac{7\pi}{4}\)

Answer:

\(\frac{\pi}{4},-\frac{7\pi}{4}\) (This is just an example answer. The actual answer depends on the position of the point on the circle. If you provide the position of the point, a more accurate answer can be given.)