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.

A

Verified step by step guidance

1. Identify the coordinates of the given point on the circle. From the image, the point is located in the first quadrant, slightly above the positive x-axis.
2. Determine the reference angle formed by the terminal side of the angle and the positive x-axis. This angle corresponds to the position of the point on the unit circle.
3. Express the first angle in radians as the reference angle itself, since it lies between 0 and 2𝜋.
4. Find the second angle by considering the angle with the same terminal side but measured in the negative (clockwise) direction. This angle will be negative and between -2𝜋 and 0.
5. Write both angles in radians, ensuring they are between -2𝜋 and 2𝜋, and verify that their terminal sides pass through the given point on the circle.

Key Concepts

Unit Circle and Coordinates

The unit circle is a circle with radius 1 centered at the origin in 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 2π radians correspond to a full circle. Angles can be positive (counterclockwise) or negative (clockwise), and multiple angles can share the same terminal side.

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.