Problem 79 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 627^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \(267^{\circ}\)

Step 1: Identify the given angle and rnage

The given angle is \(627^{\circ}\) and the required range for the coterminal angle is from \(0^{\circ}\) to \(360^{\circ}\)

Step 2: Subtract multiples of 360 degrees from the given angle

The remainder when \(627^{\circ}\) is divided by \(360^{\circ}\) will give the coterminal angle in the required range. Performing the division: \(627 \div 360 \) results in 1 remainder 267. Therefore, the coterminal angle in the required range is \(267^{\circ}\).

Key Concepts

Angle Measurement

Angles play a vital role in geometry, and they are measured in units called degrees. Each degree is a part of a full circle, which consists of 360 parts or degrees in total. To visualize how big a degree is, imagine a circle divided into 360 equal parts. Each of these parts represents a single degree. In mathematics, understanding angle measurement helps us analyze shapes, directions, and orientations. When working with angles, we often need to determine how one angle relates to another or to a given range. This concept holds particular significance when using coterminal angles, as seen in the given problem. Keeping track of the measurement unit—degrees—is essential since different scenarios may use other units like radians.

Degrees

Degrees are a system for expressing angles in a circle. The circle is divided into 360 equal segments, and each segment represents one degree. This historical division likely comes from ancient Babylonian astronomy. Notably, 360 was chosen because it is a highly composite number, offering many divisors that make calculations convenient. In terms of use, degrees simplify the understanding of angles in everyday contexts and mathematical formulas alike. Converting angles into degrees helps identify their size easily, such as deciding if one angle is greater or smaller than another. In this exercise, the angle 627° needed to be converted into a more familiar measurement by determining its coterminal angle, which is within a typical 360° range.

Angle Subtraction

When calculating coterminal angles, one often subtracts full circles or multiples of 360 degrees from a larger angle. This task simplifies the angle into a standard range of 0° to 360°, allowing us to easily compare and interpret its position relative to a complete circle. The procedure followed in the solution involved division and subtraction:

• Divide the given angle by 360 to determine how many full rotations it includes.
• Calculate the remainder, which reveals the angle measure relative to the 360° circle.

In the example, subtracting elements of a full circle from 627 degrees resulted in the remainder of 267, thus identifying it as the coterminal angle. This subtraction helps clarify the angle's position without endlessly counting rotations.

Range of Angles

Defining the range of angles is important for consistency and understanding in problem-solving. In geometry and related fields, the standard range of angles is from 0° to 360°. This method of representation makes it easier to interpret the angle's position on a circle. Finding a coterminal angle involves fitting a given angle into this accepted range. In the original exercise, identifying an angle within the 0° to 360° range was crucial for simplification. Steps to find the coterminal angle include:

• Identifying the magnitude of the given angle.
• Using division and subtraction to convert it into the 0°-360° range.

This provides an efficient way to manage angles without exceeding traditional boundaries, making them easier to work with in calculations.