Problem 18 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ 415^{\circ} $$

Step-by-Step Solution

Answer

Therefore, the measure of the angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with 415 degrees is \(55^{\circ}\).

Step 1: Identify the given angle

The angle given in this problem is 415 degrees.

Step 2: Subtract multiples of 360 degrees

Since a complete revolution corresponds to 360 degrees, subtracting multiples of 360 degrees from the given angle will not change its position on the circle. 415 degrees is more than one full circle (more than 360 degrees), subtract one 360 degrees cycle from 415 degrees: 415 - 360 = 55 degrees.

Step 3: Check the result

After subtracting 360 degrees, we end up with a 55 degrees angle, which is coterminal with 415 degrees and falls in the desired range, between \(0^{\circ}\) and \(360^{\circ}\).

Key Concepts

Angle Measurement

Understanding angle measurement is key to mastering topics such as coterminal angles. Angles can be measured in degrees, which is a common unit. The degree is a measure of rotation, where a full circle is divided into 360 equal parts. Each part is called a degree. An angle is essentially a way to measure how far one line rotates around a common point from another line. To visualize, imagine two rays originating from a point: the initial side and the terminal side. As the terminal side rotates around the initial side, the angle created is measured in degrees. For instance, if a terminal side rotates until it faces the same direction, it means it has made a full circle, corresponding to 360 degrees. Angles may also be positive or negative: - **Positive Angles**: Generated by rotating counter-clockwise from the initial position. - **Negative Angles**: Produced by rotating clockwise. Knowing that angles beyond 360 degrees or below 0 degrees can be conceptually brought back into the standard 0 to 360-degree range helps in finding equivalent or coterminal angles.

Full Rotation

A full rotation is an important concept in trigonometry and geometry. It occurs when an angle sweeps completely around a point, reaching the starting position. This full sweep is made when the terminal side of an angle rotates 360 degrees from its initial side. Understanding full rotations helps in:

• Identifying Coterminal Angles: Angles that end up in the same position after rotations can be recognized as coterminal, for instance, 60 degrees and 420 degrees (420 - 360 = 60). This is because the extra 360-degree rotation doesn't alter the terminal side's final position.
• Analyzing Periodic Functions: Most trigonometric functions have patterns that repeat every 360 degrees.

Recognizing full rotations helps simplify complex angles. When you encounter an angle like 415 degrees as in our exercise, subtract a full rotation (360 degrees). This helps find a similar smaller angle that points to the exact same position on the circle.

Degree Subtraction

Degree subtraction is a method used to simplify angles by eliminating full rotations. This is essential for finding angles within the standard range of 0 to 360 degrees, making it easier to understand and utilize them.When you have an angle greater than 360 degrees, you can subtract 360 repeatedly until the resulting angle is between 0 and 360 degrees. This process is necessary when determining coterminal angles.In our example:- Starting with a 415-degree angle.- Subtract 360 degrees: \[ 415 - 360 = 55^\circ \] The 55 degrees obtained is the angle coterminal with 415 degrees.Degree subtraction also works with negative angles. If you have a negative angle (like -45 degrees), you add 360 degrees until your angle is within the 0 to 360-degree range.