Problem 44 - Precalculus with Calculus Previews

Question

In Problems \(41-44\), find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -150^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \(210^{\circ}\).

Step 1: Understanding Coterminal Angles

An angle is coterminal with another angle if they share the same terminal side when drawn in standard position. To find a coterminal angle between \(0^{\circ}\) and \(360^{\circ}\), we can add or subtract multiples of \(360^{\circ}\) from the given angle.

Step 2: Add 360 Degrees

Since we need a positive angle between \(0^{\circ}\) and \(360^{\circ}\), add \(360^{\circ}\) to \(-150^{\circ}\) to bring it within this range:\[-150^{\circ} + 360^{\circ} = 210^{\circ}\]

Step 3: Verify the Angle

Ensure that the calculated angle \(210^{\circ}\) falls within the desired range of \(0^{\circ} \) to \(360^{\circ}\). Since \(210^{\circ}\) meets this criterion, it is the correct coterminal angle.

Key Concepts

Angle Measurement

When we talk about angle measurement, we refer to the process of determining the size of an angle in degrees or radians. Angles are measured by the amount of rotation from one ray (the initial side) to another ray (the terminal side). This rotation is typically measured in a circular path.

The unit of degrees is more commonly used for basic plane geometry. A full circle measures 360 degrees. When finding coterminal angles, an important aspect of angle measurement is ensuring that the angles are within a specified range, such as between 0° and 360° for one full rotation. This keeps the angles standardized and easier to compare.

Standard Position

An angle in standard position is set on the coordinate plane so that its vertex is at the origin \((0,0)\) and its initial side aligns with the positive x-axis. This standard positioning helps in consistently determining the measure and direction of angles.

Once the angle is placed in standard position, visualizing and calculating coterminal angles becomes more intuitive. The terminal side is where the angle "ends," and any angle that ends on this terminal side is considered coterminal. This means converting or manipulating angles within the standard position is critical for solving problems like finding coterminal angles.

Positive Angle

A positive angle is measured counterclockwise from the initial side. This movement counterclockwise is crucial for defining angles that fall within the typical range of 0° to 360°.

In contrast, a negative angle is measured clockwise from the initial side. Negative angles can be made positive by adding full rotations of 360° until the result falls within the positive range. Our task was to make the angle -150° positive by adding 360°, resulting in a positive angle of 210° within the standard 0° to 360° range.

Angle Addition

Angle addition is the process of finding coterminal angles by adding or subtracting multiples of 360 degrees. This action accounts for the fact that angles are periodic, repeating every 360°, similar to the Eart's rotation which completes a cycle every day.

In the given problem, we started with -150°. By adding 360°, a full circle, the angle becomes 210°. This process demonstrates the concept of finding coterminal angles, which remain in standard position while maintaining alignment with other possible rotations.