Problem 40 - Precalculus Mathematics for Calculus

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$361^{\circ}$$

Step-by-Step Solution

Answer

The coterminal angle is \(1^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by integer multiples of a full circle, which is \(360^{\circ}\). To find a coterminal angle between \(0^{\circ}\) and \(360^{\circ}\), we either add or subtract \(360^{\circ}\) until we fall within this range.

Step 2: Subtract 360 Degrees

Given the angle is \(361^{\circ}\), subtract \(360^{\circ}\) from it to find a coterminal angle within the desired range. \[ 361^{\circ} - 360^{\circ} = 1^{\circ} \]

Step 3: Verify the Angle

The calculated angle \(1^{\circ}\) is within the range of \(0^{\circ}\) to \(360^{\circ}\), which confirms that \(1^{\circ}\) is the correct coterminal angle.

Key Concepts

Angle Measurement

When discussing angles, it's crucial to understand how they are measured. Angle measurement allows us to compare and communicate the size of angles in mathematics and everyday situations easily.

To measure an angle, we need two rays (or lines) that share a common endpoint known as the vertex. The space or rotation between these lines is what we refer to as the angle. The size of this angle is determined by how much one line is rotated with respect to the other.

We often use units like degrees or radians to describe the magnitude of an angle, with degrees being the more common choice in many fields. Understanding how angles are measured is fundamental when working with geometric concepts like coterminal angles.

Degree

The degree is a unit for measuring angles, forming the basis of most angle-related problems and solutions. A full circle is divided into 360 equal parts, and each part is one degree. This division into 360 is historically based on Babylonian mathematics, which used a sexagesimal (base 60) system.

Here are some quick facts about degrees:

• A right angle is exactly 90 degrees.
• A straight angle (or half-circle) measures 180 degrees.
• Full circles encompass 360 degrees.

Degrees provide a simple and effective way to express angles, especially when dealing with geometric shapes and rotation. This is why you often hear about angles in degrees when discussing coterminal angles, which can be adjusted by adding or subtracting multiples of a full circle, or 360 degrees, to find their counterparts.

Full Circle

In the realm of angles, the term "full circle" is used to describe a complete rotation around a central point, covering 360 degrees. This concept is essential when dealing with coterminal angles because it allows us to find different angles that visually coincide using rotations.

Understanding a full circle in terms of degrees can be visualized easily on a clock face:

• Moving clockwise from 12 to 12 again completes a full rotation, or 360 degrees.
• If you move from 1 to 1, you've simply added another 30-degree increment.

This full-circle logic can also be used to determine angles that share the same terminal side. When finding coterminal angles between 0 and 360 degrees, subtracting or adding multiples of 360 helps pinpoint the angle that completes these rotations without exceeding the bounds of a single circle.