Problem 16 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ 500^{\circ} $$

Step-by-Step Solution

Answer

The angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(500^{\circ}\) is \(140^{\circ}\).

Step 1: Identify the Problem

The problem is to find an angle that is coterminal with the given angle of \(500^{\circ}\), but falls within the range of \(0^{\circ}\) to \(360^{\circ}\). So, The strategy to solve this problem will be adding or subtracting multiples of \(360^{\circ}\) from the given angle.

Step 2: Subtracting Multiples of 360

To get the angle within the desired range, subtract \(360^{\circ}\) from the given angle \(500^{\circ}\) until the result falls within \(0^{\circ}\) to \(360^{\circ}\). If we do this subtraction once, we get \(500^{\circ} - 360^{\circ} = 140^{\circ}\).

Step 3: Check the answer

Check if the resulting angle of \(140^{\circ}\) falls within the range \(0^{\circ}\) to \(360^{\circ}\). As we can see that it does fall within this range, this is the required coterminal angle.

Key Concepts

Understanding Angle Measurement

Angles are often measured in degrees, where a full circle is made up of 360 degrees. Each degree is a small part of the circle's circumference. Angles can be larger than 360 degrees or even negative. In these cases, they may complete one or more full rotations around a circle.

This is where the concept of coterminal angles comes into play. Coterminal angles have the same initial and terminal sides, meaning they look the same on a circle but can have different measures. For example, an angle of 500 degrees appears the same as a 140-degree angle because both end at the same spot after one or more full circle rotation(s).

Understanding angle measurement, especially in the context of coterminal angles, is fundamental for solving many problems in geometry and trigonometry. To identify coterminal angles, we use arithmetic operations to adjust the angle until it fits within a specific range, often 0 to 360 degrees.

Subtracting Multiples for Coterminal Angles

When finding a coterminal angle that fits within a specific range, such as 0 to 360 degrees, it's useful to adjust the angle by adding or subtracting full circles. Each full circle accounts for 360 degrees.

To find a coterminal angle for a measure like 500 degrees, first check if subtracting 360 degrees places it within the desired range. With the original angle of 500 degrees, subtracting 360 degrees just once gives us:

• Subtraction Step: \[500^{\circ} - 360^{\circ} = 140^{\circ}\]

The result here, 140 degrees, conveniently falls within the range of 0 to 360 degrees. Remember, if the result after one subtraction doesn't fit, continue subtracting 360 as needed! This process effectively brings overly large angles back to the more familiar and manageable initial circle.

The Role of Angle Range in Geometry

The angle range is crucial when solving problems with coterminal angles. Often, we aim to convert whatever angle is given into an equivalent angle that lies within a typical and easy-to-interpret range, like 0 to 360 degrees. This makes calculations simpler and interpretations straightforward.

If an angle is expressed as something larger than 360 degrees, it implies more than a full rotation around the circle. Similarly, if it's negative, it indicates a clockwise direction from the initial position. In formal solutions, converting to a standard range helps align with conventional visualization and usage in geometric and trigonometric practices. In the case of our example, adjusting 500 degrees into the 0 to 360-degree range simplifies comparisons, interpretations, and further computations. The final angle of 140 degrees clearly fits this range, confirming the transformation process is complete and correct.