Problem 19 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ -180^{\circ} $$

Step-by-Step Solution

Answer

The measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-180^{\circ}\) is \(180^{\circ}\).

Step 1: Understanding the Problem

The goal here is to find an angle coterminal to \(-180^{\circ}\) that lies within the \(0^{\circ}\) and \(360^{\circ}\) range. To find a coterminal angle in the positive direction, we simply add \(360^{\circ}\) to the given angle. This is due to the fact that a complete circular rotation equals \(360^{\circ}\). If the original angle is negative, we add \(360^{\circ}\) until reaching a degree that is between \(0^{\circ}\) and \(360^{\circ}\).

Step 2: Finding Coterminal Angle

Given the angle \(-180^{\circ}\), we start adding \(360^{\circ}\) to it: \(-180^{\circ} + 360^{\circ} = 180^{\circ}\)

Step 3: Confirming the Outcome

We now have the angle \(180^{\circ}\), which lies between \(0^{\circ}\) and \(360^{\circ}\). Hence, we have found the angle coterminal with \(-180^{\circ}\).

Key Concepts

Circle Rotation

Understanding the concept of circle rotation is key to grasping why adding or subtracting specific amounts from angles sometimes does not change their physical representation on a plane.

• Essentially, a full circle measures 360 degrees. It's like completing a lap around a track.
• When we talk about angles and their measures, rotating a full 360 degrees means one complete rotation, bringing you back to your starting position.

Coterminal angles occur when an angle has undergone one or more full rotations, but still appears visually as the same angle. For instance, if you start at a 0-degree angle and rotate 360 degrees, you end up at 0 degrees again. Similarly, if you add or subtract any multiple of 360 degrees from an angle, the result is a coterminal angle. This means it points in the same direction, even if it seems like a different measure due to its multiple rounds around the circle. To sum it up, circle rotation helps us understand that angles repeating every 360 degrees end up pointing in the same direction. When asked to find a coterminal angle like in our exercise, you simply adjust the original angle by full circles until it falls within the specified range, in this case, from 0 to 360 degrees.

Angle Measures

Angle measures are how we evaluate how far one line has rotated about a point from another line, usually expressed in degrees. It's like measuring how wide open a door is from a specific starting point.

• Standard measurement runs from 0 to 360 degrees, which corresponds to a complete circle around the point.
• When dealing with angles in a coordinate system, you'll often mark angles from the positive x-axis.

Each crucial measure on the circle, like the problem of finding a coterminal angle for -180 degrees, involves using this 360-degree system. By continuously adding or subtracting 360 degrees, we can convert any given angle into its corresponding coterminal angle. This allows it to fit neatly into our 0 to 360-degree understanding of a circle, like resolving negative angles where such standard measures don't naturally fall. In the original problem, we found an angle coterminal to -180 degrees by adding 360 degrees, resulting in an equivalent angle of 180 degrees — well-fitted into our natural 0 to 360-degree range.

Positive and Negative Angles

Positive and negative angles describe direction in which rotation occurs. Getting this is crucial for solving questions about coterminal angles.

• A positive angle is one that is measured counterclockwise from the positive x-axis towards the positive y-axis.
• Conversely, a negative angle moves clockwise from the positive x-axis towards the negative y-axis.

In our case, the original angle given was -180 degrees. Negative here indicates that it rotates clockwise halfway around the circle. With this information, to make it a positive angle within our reference of 0 to 360 degrees, we added 360 degrees; hence the angle's direction is rotated back half a circle, resulting in the positive angle of 180 degrees. This conversion from negative to positive rotation is possible, thanks to understanding angle direction, by rotating around the circle incrementally (like we add 360 degrees here), aligning the angle measurement to a positive direction in degree terms spanning 0 to 360 degrees.