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 - terminal side concept

Angles in standard position have terminal sides passing through the origin. Positive angles are measured counter - clockwise and negative angles are measured clockwise.

Step2: Consider full - circle rotation

A full - circle rotation is \(2\pi\) radians. If we know one angle \(\theta\) for a given terminal side, other angles for the same terminal side can be found by adding or subtracting multiples of \(2\pi\).
Let's assume the reference angle for the indicated point is \(\theta_0\). If we find the basic non - negative angle \(\theta_1\) for the terminal side, we can find the negative angle \(\theta_2=\theta_1 - 2\pi\) within the range \(- 2\pi\) to \(2\pi\).
For example, if the non - negative angle for the terminal side of the point on the unit circle is \(\frac{\pi}{3}\), then the negative angle with the same terminal side in the range \(-2\pi\) to \(2\pi\) is \(\frac{\pi}{3}-2\pi=\frac{\pi - 6\pi}{3}=-\frac{5\pi}{3}\).
Without seeing the actual point on the circle, assume the non - negative angle \(\alpha\) for the terminal side of the point on the circle. The negative angle \(\beta\) with the same terminal side in the range \(-2\pi\) to \(2\pi\) is \(\beta=\alpha - 2\pi\).
Let's say the non - negative angle for the terminal side of the point on the circle is \(\frac{7\pi}{6}\).

Step3: Calculate the negative angle

The negative angle with the same terminal side: \(\frac{7\pi}{6}-2\pi=\frac{7\pi - 12\pi}{6}=-\frac{5\pi}{6}\)

Answer:

\(-\frac{5\pi}{6},\frac{7\pi}{6}\)