Problem 24 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ -500^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \(220^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They differ by full rotations (multiples of 360°). Given an angle, we can find a coterminal angle by adding or subtracting 360° until the angle lies within the specified range.

Step 2: Adding 360° Until the Range is Met

We start with the angle \(-500^{\circ}\),and we need to add\(360^{\circ}\)repeatedly until \(0 \leq \theta < 360\). First addition:\(-500^{\circ} + 360^{\circ} = -140^{\circ}\). This is not yet within the range (0, 360), so we add again:\(-140^{\circ} + 360^{\circ} = 220^{\circ}\). Now, \(220^{\circ}\) is within the required range.

Key Concepts

Angle Measurement

Understanding angle measurement is crucial in geometry and trigonometry. Angles are measured in degrees, which represent the amount of rotation from one arm of the angle to the other. A full rotation around a circle is 360 degrees, and this is the foundation for many angle-related concepts.

When working with angles, it's important to recognize that several angles can share the same terminal side. These are known as coterminal angles. They may look different numerically, but visually they point in the same direction. To find coterminal angles, you can add or subtract multiples of 360 degrees. This helps you identify angles that represent the same rotation in practical terms.

Negative Angles

Negative angles might seem intimidating at first, but they are a natural extension of the concept of rotation. While positive angles denote a counterclockwise rotation from the positive x-axis, negative angles indicate a clockwise rotation.

When dealing with angle problems, you can convert a negative angle to a positive coterminal angle by adding 360 degrees. For example, consider an angle of -500°, which initially points in a clockwise direction. By adding 360 degrees repeatedly, you transform it into a positive angle within the range of 0° to 360°.

• This conversion is helpful as it allows you to visualize the angle on a standard unit circle and simplifies calculations.
• Each addition of 360 degrees represents a full circle rotation, gradually bringing the angle into the desired range.

360 Degree Cycle

The 360 degree cycle is an essential concept in understanding how angles and rotations work. A circle has 360 degrees, and any rotation around a circle that exceeds 360 degrees is just an extension of this complete revolution, either in the positive or negative direction.

Think of a 360 degree cycle like the hand of a clock moving around the face. Once it completes a full cycle, it starts over again from the same point. This means:

• Additions or subtractions of 360 degrees to an angle, whether it is initially negative or positive, doesn't change the direction it points to on the circle.
• For example, moving from -500° to -140° and finally to 220° just involves resetting the clock a few times until the hands rest at a familiar time.

Understanding this cycle can simplify how you approach angle measurement, making it easier to find coterminal angles and work with rotations.