Problem 73 - 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. $$-390^{\circ}$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle is 330°.

Step 1: Understand the concept of coterminal angles

Two angles are coterminal if they differ by a whole number of full rotations. In degree measure, one full rotation is 360 degrees. Therefore, to find a coterminal angle, you can either add or subtract multiples of 360° from the given angle.

Step 2: Add 360° incrementally to find a positive angle

Given the angle of -390°: Add 360° to the angle until the result is positive: -390° + 360° = -30° -30° is still negative, so add 360° again: -30° + 360° = 330° Now, 330° is a positive angle.

Step 3: Ensure the angle is the smallest positive measure

The result of the previous step is 330°, which is the smallest positive measure because any further addition of 360° would result in an angle greater than 360°, which we want to avoid.

Key Concepts

Angle Measure

Angles are a fundamental part of mathematics and are measured in degrees or radians. An angle is formed by two rays sharing a common endpoint, known as the vertex of the angle. The opening between these two rays defines the size of the angle. The measure of an angle can be thought of as how much one ray needs to rotate about the vertex to overlap with the other ray.
The angle measure tells us about the rotation needed. This measure is crucial for understanding and defining positions and changes in direction.
In everyday life and in various branches of science and engineering, angle measurement is vital.

• In degrees, a complete rotation around a circle is 360°.
• In radians, a full rotation is equivalent to \(2\pi\) radians.

Keeping track of the angle measure is key when determining if two angles are coterminal, as they differ by full circle rotations.

Degree Measure

The degree is a common unit used to measure angles. One degree represents \(\frac{1}{360}\) of a full circle. This means that when you travel around a full circle, you travel through 360 degrees.
Measuring in degrees allows simpler calculations and easy understanding in many situations. For instance, constructing angles with precise degree measures is typical in architecture and navigation.
To find coterminal angles using degree measure, you can add or subtract multiples of 360°, since 360° is one complete rotation. This means:

• If an angle is \(-450°\), adding \(360°\) repeatedly till you reach a positive angle involves:
• \(-450° + 360° = -90°\) (still negative)
• \(-90° + 360° = 270°\) (now positive and coterminal)

This method ensures that the angle remains within the standard 0° to 360° range, facilitating simpler applications and interpretations.

Full Rotation

A full rotation in degrees means turning around in a complete circle, which is 360 degrees. It is like spinning once completely back to where you started.
Understanding full rotation helps with finding coterminal angles. Coterminal angles are those that end at the same position in their rotation. This means:

• Angles such as 30°, 390° (which is 30° + 360°), or -330° (which is 30° - 360°), are coterminal.

In problems with negative angles, you sometimes perform these full rotations in reverse by adding 360° to reach the smallest positive angle.
Full rotation concepts help to reset the angle measurement, allowing for consistent understanding and application in various fields such as physics, engineering, and computer graphics.