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.

D

Verified step by step guidance

1. Identify the position of point D on the unit circle. From the image, point D is located in the second quadrant, slightly above the horizontal axis.
2. Determine the reference angle for point D. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. Count the tick marks from the positive x-axis to point D to estimate this angle in terms of \( \pi \).
3. Express the first angle \( \theta_1 \) in radians between 0 and 2\( \pi \) that corresponds to point D. Since point D is in the second quadrant, \( \theta_1 = \pi - \text{reference angle} \).
4. Find the second angle \( \theta_2 \) between -2\( \pi \) and 0 that has the same terminal side as point D. This angle can be found by subtracting 2\( \pi \) from \( \theta_1 \), so \( \theta_2 = \theta_1 - 2\pi \).
5. Verify that both angles \( \theta_1 \) and \( \theta_2 \) have terminal sides passing through point D by considering their positions on the unit circle.

Key Concepts

Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. 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 one full rotation equals 2π radians. Negative angles represent clockwise rotation, and positive angles represent counterclockwise rotation.

Finding Angles with the Same Terminal Side

Two angles have the same terminal side if they differ by a full rotation of 2π radians. To find two angles between -2π and 2π passing through a given point on the unit circle, identify the reference angle and then add or subtract multiples of 2π to find the corresponding angles within the specified range.