Problem 26 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ 360^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is 0 degrees.

Step 1: Understand the Problem

A coterminal angle is an angle that shares the same initial and terminal sides, but may have a different measure. We need to find such an angle for the given angle that falls within the interval of 0 degrees to less than 360 degrees.

Step 2: Identify Provided Angle and Adjustment Needs

The provided angle is 360 degrees. Since coterminal angles differ by full rotations of 360 degrees, we need to adjust this angle to fit into the specified interval: \(0 \leq \theta < 360\).

Step 3: Calculate the Coterminal Angle

Subtract 360 degrees from the given angle since 360 is exactly one full rotation: \(360 - 360 = 0\).

Step 4: Verify the Angle Falls in the Interval

Check if the computed coterminal angle (0 degrees) satisfies the condition \(0 \leq \theta < 360\). Since it does, the coterminal angle is 0 degrees.

Key Concepts

Angle Measurement

Angles are fundamental in geometry, helping describe the rotation between two rays sharing a common endpoint. The endpoint, also known as the vertex, is the point where the two rays meet. To measure an angle, we determine the amount of rotation from one ray to the other. This measurement can be expressed in different units, such as degrees or radians, depending on the context. When we discuss angle measurement in degrees, we often think in a circular context. A full circle is 360 degrees, meaning if you rotate around a point once, you've completed a 360-degree rotation. Dividing a circle into 360 parts makes each degree a tiny part of the entire whole, allowing us to measure angles from 0 to 360 degrees. For angles greater than 360 or negative angles, we use coterminal angles to keep them within the standard range.

Degrees

Degrees are the most commonly used unit for measuring angles, particularly in everyday applications and schooling. Simply put, when we talk about an angle's measure, we're referring to the amount of turn between two arms or sides using degrees as our unit.

• One full rotation around a circle is 360 degrees.
• A right angle, which forms an "L" shape, is 90 degrees.
• A straight angle, which looks like a straight line, measures 180 degrees.

It's crucial to understand these basic measurements to complete more complex problems involving angles. Degrees allow us to easily partition and describe the space within circles and around points. When given angles, we often need to find equivalent measurements (coterminal angles) to solve problems efficiently.

Rotation in Angles

Rotation is the movement of an angle's initial side to its terminal side, creating a circular movement. This idea of rotation is how we understand angles as representing turns or spins. Every full rotation around a circle brings us back to our starting point, completing a cycle of 360 degrees. Coterminal angles are closely tied to this concept of rotation. Two angles are considered coterminal if the amount of rotation they represent results in the same position. Here’s how you can determine coterminal angles:

• Add or subtract multiples of 360 degrees to find a coterminal angle within a desired range.
• For example, to find a coterminal angle less than 360 degrees for 720 degrees, subtract 360 twice to get to 0 degrees.

When working with coterminal angles, always ensure the adjusted angle fits within the standard interval (commonly 0 to 360 degrees) to avoid confusion and to standardize measurements across various problems.