Problem 13 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(60^{\circ}\)

Step-by-Step Solution

Answer

Positive: 420°, Negative: -300°.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They can be found by adding or subtracting full rotations, which are multiples of 360° for degrees.

Step 2: Finding a Positive Coterminal Angle

To find a positive coterminal angle for 60°, add 360° to it: \[60° + 360° = 420°\].Thus, 420° is a positive coterminal angle.

Step 3: Finding a Negative Coterminal Angle

To find a negative coterminal angle for 60°, subtract 360° from it: \[60° - 360° = -300°\].Thus, -300°is a negative coterminal angle.

Key Concepts

Positive Angle

A positive angle is one that measures in a counterclockwise direction from the initial side to the terminal side. This direction follows the pattern used in mathematics and physics. Picture the hands of a clock: when you move against the usual clockwise direction, you get a positive angle.

• Positive angles begin at 0° and increase towards 360°.
• They are most commonly used in problems relating to rotation and trigonometry.

For example, when finding a coterminal angle to 60° that is positive, you would add full rotations of 360° until you have the desired angle. Here, one full rotation added to 60° is 420°, still positive and coterminal.

Negative Angle

Negative angles measure in the opposite direction, clockwise from the initial side to the terminal side. Think of it as reversing the clock hands' regular movement.

• Negative angles count counterclockwise starting from 0° and decrease.
• They are useful to describe rotations that do not follow the normal positive path.

In reaching a negative coterminal angle for 60°, we subtract a full loop of 360°. The result is -300°, which represents the same direction but in a backward path compared to the positive angle.

Angle Measurement

Understanding angle measurement is essential in mathematics to describe rotation. An angle is formed by two rays (sides) sharing an endpoint, known as the vertex. The amount of turn between each ray indicates the angle's magnitude.

• Angles can be measured in degrees, radians, or other units.
• Degrees use a full circle of 360 units, while radians are based on the circle's radius.

When identifying coterminal angles, it's critical to know how angles can be expressed in different ranges such as 0° to 360° or negative to positive values.

Degrees

Degrees are the most common form of expressing angle measures. They divide a circle into 360 parts, allowing for precise determination of an angle's size.

• Each degree can be further divided into 60 minutes and each minute into 60 seconds.
• This system helps with fine precision and detailed angle measurements.

In trigonometry, understanding degrees is crucial to solving problems related to circles and rotations. The need to find coterminal angles often revolves around adding or subtracting these 360-degree values.

Full Rotations

A full rotation is a complete turn around a circle, equating to 360 degrees. It's the fundamental building block in calculating coterminal angles.

• Each full rotation resets the angle back to 0°, ready to begin a new turn.
• Multiple full rotations can be summed to provide angles over 360° or into negatives below 0°.

The concept of full rotations is key when assessing and establishing coterminal angles. It helps in understanding how angles like 420° and -300° can point in the same direction as 60° through the lens of multiple or reversed rotations.