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 with terminal side passing through origin and a point on unit - circle can be found by considering positive and negative rotations.

Step2: Assume a general approach

Let's assume the circle is a unit - circle. If we know the position of the point on the circle, we can find the angles. For example, if the point is on the positive x - axis, the angles are \(0\) and \(- 2\pi\). If the point is on the negative x - axis, the angles are \(\pi\) and \(-\pi\). Since the problem doesn't show the point, we'll use the general rule that angles \(\theta\) and \(\theta\pm2k\pi\) (\(k\in\mathbb{Z}\)) have the same terminal side. We want angles in the range \(-2\pi<\theta < 2\pi\).
Let the reference angle be \(\alpha\). The two angles \(\theta_1=\alpha\) and \(\theta_2=\alpha - 2\pi\) (if \(\alpha>0\)) or \(\theta_2=\alpha + 2\pi\) (if \(\alpha<0\)) will be in the required range.
Suppose the point on the circle corresponds to an angle \(\alpha\) in the first - quadrant. Then the two angles are \(\alpha\) and \(\alpha - 2\pi\).

Answer:

Since the point on the circle is not given, we cannot provide specific numerical answers. But the general form of the two angles \(\theta_1\) and \(\theta_2\) such that \(-2\pi<\theta_1,\theta_2<2\pi\) and they have the same terminal side passing through the origin and the given point on the circle are: if the reference angle is \(\alpha\), then \(\theta_1=\alpha\) (where \(-2\pi<\alpha<2\pi\)) and \(\theta_2=\alpha\pm2\pi\) adjusted to be in the range \(-2\pi <\theta_2<2\pi\).