Problem 82 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -609^{\circ} $$

Step-by-Step Solution

Answer

The measure of an angle between 0 degrees and 360 degrees that is coterminal with -609 degrees is 111 degrees.

Step 1: Identify given angle

The given angle is -609 degrees.

Step 2: Start adding complete 360 degree revolutions

While the angle is negative, start adding 360. -609 + 360 = -249. It's still negative, so add another 360. -249 + 360 = 111.

Step 3: Validate result

The result, 111 degrees, is positive and between 0 and 360, so it's in the correct range.

Key Concepts

Angle Measurement

Understanding angle measurement is crucial for grasping the idea of coterminal angles. Angles are typically measured in degrees, which is a way to describe the amount of rotation from the initial side (starting position) of the angle to its terminal side (ending position). A full circle measures 360 degrees, making it one complete rotation.
Angles can be positive, indicating a counter-clockwise rotation, or negative, indicating a clockwise rotation. An angle is measured starting from the positive x-axis, which is the usual reference point in coordinate geometry.
When asked to find coterminal angles, the goal is to identify angles that share the same initial and terminal sides. This often involves adding or subtracting full rotations (360 degrees) to reach a measurement within a desired range, typically between 0 and 360 degrees.

Negative Angles

Negative angles can initially be confusing but understanding them is key to solving problems involving rotations. Unlike positive angles, which are measured in a counter-clockwise direction, negative angles are measured clockwise. This means that if you rotate -90 degrees, you’re actually moving 90 degrees clockwise from the starting line.
The concept of negative angles is essential in finding coterminal angles. Beginning with the given negative angle, it's common practice to continuously add 360 degrees (one full rotation) until the angle becomes positive. This method helps you transform a negative measure into a positive angle that falls within the standard 0 to 360-degree range.
For example, starting from -609 degrees, adding 360 degrees yielded -249 degrees, and adding another 360 resulted in 111 degrees, a positive angle between 0 and 360 degrees.

360 Degree Rotation

A 360-degree rotation refers to a complete circular turn. This concept is fundamental when working with coterminal angles because it enables you to convert any angle into one within the standard 0 to 360-degree range. By adding or subtracting multiples of 360, you align any angle to have the same terminal side as those within a single full rotation.
In the context of the original problem, dealing with the angle -609 degrees required successive additions of 360 to reach a valid angle within this range.
Whenever you have a problem involving coterminal angles, repeatedly adding or subtracting 360 degrees is a reliable way to adjust the angle measurement to fit the desired interval, ensuring it makes sense in practical applications and aligns well with problems in geometry and trigonometry.