Problem 14 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ -405^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle of -405 degrees that lies between 0 and 360 degrees is 315 degrees.

Step 1: Understanding Coterminal Angles

Two angles are said to be coterminal if they share the same terminal side. As per a complete rotation in a circle, an angle is equivalent to an angle plus any integer multiple of 360 degrees.

Step 2: Determine the Original Angle

The original angle given is -405 degrees.

Step 3: Calculate Coterminal Angle

To find the positive coterminal angle between 0 and 360 degrees for -405 degrees, 360 degrees can be added to the original angle. \[-405^{\circ} + 360^{\circ} = -45^{\circ}\] As the resulting angle of -45 degrees is not within the desired range (0 to 360), another 360 must be added, resulting in \[-45^{\circ} + 360^{\circ} = 315^{\circ} \] 315 degrees is within the specified range, therefore it is the angle needed.

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side, meaning they end in the same position once drawn on a coordinate plane. It doesn't matter how many times you've gone around the circle, just where you finally end up. To find multiple coterminal angles of a given angle, you can keep adding or subtracting full rotations of \(360^{\circ}\). For example:

• Adding \(360^{\circ}\) to an angle: \( \theta + 360^{\circ} \)
• Subtracting \(360^{\circ}\) from an angle: \( \theta - 360^{\circ} \)

This process can be repeated until the angle is within the desired range, such as between \(0^{\circ}\) and \(360^{\circ}\). This way, you ensure the angle matches a standard position.

Rotation

Rotation is a fundamental concept when working with angles and coterminal angles. A full rotation around a circle is \(360^{\circ}\). When an angle measurement exceeds a full rotation, it means we have circled the coordinate system more than once. Rotation can occur in the following ways:

• Counterclockwise: The traditional direction for positive angles.
• Clockwise: Used for negative angles. Each \(360^{\circ}\) is considered as a full rotation.

By understanding rotation, you can easily differentiate between positive and negative angles and identify how they can be expressed as coterminal angles.

Measurement of Angles

To measure angles, we use degrees, which divide a full rotation into 360 equal parts. The degree system is used universally for various geometrical calculations and is vital in identifying coterminal angles. Here's how you can measure angles efficiently:

• An angle is expressed depending on its position relative to another direction, often horizontal or vertical.
• The starting point of an angle is known as the initial side, and the ending point is the terminal side.

Whenever you're determining which angle is coterminal, you'd be looking for the angle measurement that rests between \(0^{\circ}\) and \(360^{\circ}\). This ensures understanding and uniformity across mathematical and physical applications.