Problem 79 - Precalculus

Question

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$-30$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle of -30° is 330° .

Step 1: Understand the Concept of Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle, you can add or subtract full rotations (360°) until the angle is within the desired range, typically between 0° and 360°.

Step 2: Identify the Given Angle

The given angle is -30° . Since the angle is negative, it means that it is measured clockwise from the positive x-axis.

Step 3: Make the Angle Positive

To find the smallest positive coterminal angle, add 360° (a full rotation) to the given angle. \( -30° + 360° = 330° \).

Step 4: Verify the Result

330° is a positive angle, and since it is less than 360°, it is within the typical range for positive angles. It represents the smallest positive angle coterminal with the original angle, -30° .

Key Concepts

Angle Measurement

Angle measurement is crucial in understanding how angles are determined and represented. Angles are typically measured in degrees or radians. One full circle around a point consists of 360 degrees or can be represented as \(2\pi\) radians.
For everyday situations, degrees are often used because they provide a more intuitive feel for the part of a circle the angle represents.

• A 90-degree angle corresponds to a quarter turn.
• 180 degrees represents a half turn.
• 360 degrees is a full rotation.

When working with coterminal angles, knowing these measurements helps in quickly identifying angles that we want to describe or analyze. To switch from negative to a positive angle measure, you can add 360 degrees to ensure we're within the standard circle range. This concept of angle measurement is directly applied when finding coterminal angles, as seen in exercises like the one given with an angle of -30 degrees.

Positive Angle

A positive angle is an angle measured counterclockwise from the positive x-axis. In many real-life and mathematical contexts, working with positive angles simplifies calculations and interpretations.
When you have a negative angle, such as \(-30°\), it's expressing a clockwise rotation. To convert this into a positive angle:

• Add 360° to bring it back within the usual positive range.

This ensures the angle still accurately represents the same position around the circle. For instance, the calculation of \(-30° + 360°\) results in \(330°\).
This angle exhibits a concise and clear representation of the rotational position while maintaining a positive angle measure.
Usually, positive angles are easier to work with as they align with the standard mathematical conventions, making the interpretation straightforward in many applications.

Standard Position

Standard position is a foundational concept in understanding angles, especially in trigonometry and geometry.
An angle is said to be in standard position when its vertex is located at the origin of a coordinate plane, and its initial side lies along the positive x-axis. From this position, angles are measured by the rotation of the terminal side:

• Counterclockwise rotation corresponds to a positive angle.
• Clockwise rotation results in a negative angle.

When determining coterminal angles, the begin point of reference is always the standard position for the angle. Considering the exercise's original angle \(-30°\) in standard position: adding 360 degrees positions the angle at \(330°\), keeping it coterminal with the original yet situated within the preferred positive range.
This standardized method helps in systematically understanding and calculating angles within their respective coordinate systems.