Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.
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Verified step by step guidance

1. Identify the coordinates of the given point on the circle. Here, the point K is located at the top of the circle on the positive y-axis, so its coordinates are (0, 1).
2. Recall that the angle in standard position is measured from the positive x-axis counterclockwise. Since the point is on the positive y-axis, the reference angle is \( \frac{\pi}{2} \).
3. Find the two angles between \(-2\pi\) and \(2\pi\) whose terminal sides pass through the point (0, 1). One angle is the positive angle \( \frac{\pi}{2} \).
4. The other angle is the negative angle that corresponds to the same terminal side, which is found by subtracting \( 2\pi \) from \( \frac{\pi}{2} \), giving \( \frac{\pi}{2} - 2\pi = -\frac{3\pi}{2} \).
5. Thus, the two angles in radians between \(-2\pi\) and \(2\pi\) are \( \frac{\pi}{2} \) and \( -\frac{3\pi}{2} \).

Key Concepts

Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin of the coordinate system. Points on the unit circle correspond to angles measured from the positive x-axis, and their coordinates (x, y) represent the cosine and sine of those angles, respectively.

Angles in Standard Position and Radians

An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. Angles are measured in radians, where 2π radians correspond to a full circle. Negative angles represent clockwise rotation, and positive angles represent counterclockwise rotation.

Finding Coterminal Angles

Coterminal angles share the same terminal side but differ by full rotations of 2π radians. To find two angles between -2π and 2π with the same terminal side, add or subtract multiples of 2π from a given angle, ensuring the angles fall within the specified range.