Problem 77 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -225^{\circ} $$

Step-by-Step Solution

Answer

The measure of the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-225^{\circ}\) is \(135^{\circ}\).

Step 1: Recognize what a coterminal angle is

By definition, an angle that is coterminal with another can be found by adding or subtracting multiples of 360 degrees. The goal here is to add 360 degrees until the result is greater than 0 degrees and less than 360 degrees.

Step 2: Finding the Coterminal Angle

Given that the initial angle is -225 degrees, add 360 degrees to get the positive coterminal angle. This is done using the equation (angle measure) + n(360) where n can be any integer, the task requires n=1. So, -225 + 1(360) = 135 degrees.

Key Concepts

Angle Measurement

When we talk about angle measurement, we're looking at the size of an angle created by two rays sharing a common endpoint, known as a vertex. Angles can be measured in degrees, which is the most common unit. A full rotation around a circle corresponds to 360 degrees. Important points on this circle include:

• 0 degrees, which is the starting point for angle measurement.
• 90 degrees, which is a right angle.
• 180 degrees, which forms a straight line.
• 270 degrees, representing three quarters of a full circle.
• 360 degrees, completing the full circle.

We use degrees because they divide the circle into 360 equal parts, making them quite manageable for describing rotations and directions. In our exercise, we deal with angles that form as we rotate either clockwise or counterclockwise from the 0 degree mark.

Degree System

The degree system is a framework for measuring and categorizing angles based on the circle's division into 360 parts. This system not only helps us measure angles but also allows us to identify specific positions and intervals on a circle. A crucial aspect of this system is the understanding of coterminal angles. Coterminal angles are angles that share the same terminal side when drawn in standard position on a coordinate plane. To find coterminal angles, you can add or subtract multiples of 360 degrees to any given angle. For example, if you have an angle of -225 degrees like in our problem, you can keep adding 360 degrees to it until you get an angle within the typical range of 0 to 360 degrees.
In this case, by adding 360 degrees to -225 degrees, we reach the angle of 135 degrees, which is between 0 and 360 and is coterminal with -225 degrees.

Positive and Negative Angles

Angles can have both positive and negative measurements, and understanding this concept is key to working with coterminal angles. Positive angles represent counterclockwise rotations from the starting line (known as the positive x-axis) when drawn in standard position. Conversely, negative angles depict clockwise rotations.
It's essential to realize that the same position on a circle can be represented by both a positive or negative angle based on the direction of rotation. For instance, -225 degrees and 135 degrees describe the same position, but their signs indicate different rotational paths: -225 degrees results from rotating 225 degrees clockwise, while 135 degrees comes from rotating 135 degrees counterclockwise.
By converting negative angles to positive ones, we stay within the preferred standard range (0 to 360 degrees) for ease of interpretation and calculation.