Problem 17 - Algebra 2

Question

Find the measure of an angle between \(0^{\circ}\) and \(360^{\circ}\) coterminal with each given angle. $$ -210^{\circ} $$

Step-by-Step Solution

Answer

The angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-210^{\circ}\) is \(150^{\circ}\).

Step 1: Understanding the Problem

The given angle is \(-210^{\circ}\). The objective is to find an angle that is coterminal with the given one and lies between \(0^{\circ}\) and \(360^{\circ}\). This can be accomplished by adding or subtracting multiples of \(360^{\circ}\) until the resultant angle falls within the target range.

Step 2: Adding 360 Degrees to the Angle

To find a positive coterminal angle, simply add \(360^{\circ}\) to the given angle. So, -210 + 360 = \(150^{\circ}\).

Step 3: Verifying the Solution

Now that the angle found is \(150^{\circ}\), it can be checked whether it lies between \(0^{\circ}\) and \(360^{\circ}\). Indeed, \(150^{\circ}\) does fall within that range, indicating that this is the coterminal angle we were seeking.

Key Concepts

Understanding Angle Measurement

Angle measurement is a crucial component in understanding geometry and trigonometry. An angle is essentially the amount of rotation required to superimpose one of two intersecting lines or planes on the other. These measurements are usually given in degrees.
Degrees are the most commonly used unit for measuring angles.
Every complete rotation around a single point is equal to 360 degrees. When you encounter angles outside this range, understanding how to convert them into a more useful form is important for solving problems.

Everything You Need to Know About Degrees

Degrees describe how much of a full circle an angle covers. A full circle is made up of 360 degrees. This makes it easy to think about angles as part of the circle.
By using degrees, you can divide this circle into different units:

• 90 degrees, which forms a right angle (a quarter of a circle).
• 180 degrees, which forms a straight angle (half a circle).
• 270 degrees, forming three-quarters of a circle.
• 360 degrees, completing the full circle.

This method is straightforward for solving many geometry problems, because it matches our intuitive understanding of rotations, such as a clock's hands moving around the dial.
Understanding degrees is essential for determining angles that could be outside the standard 0 to 360-degree range.

Algebraic Problem Solving for Coterminal Angles

Solving for coterminal angles is an algebraic problem-solving exercise that often involves simple addition or subtraction.
If you are given an angle like -210 degrees, and need to find a coterminal angle between 0 and 360 degrees, you can add multiples of 360 degrees until you get a positive result within that range.

• For the angle -210 degrees, you add 360 degrees and get 150 degrees.
• Verification is important to ensure your result lies within the desired range of 0 to 360 degrees.

This sort of technique can also be used to solve problems involving angles in the positive direction, where subtracting 360 degrees can bring an angle back to the desired range.
Learning how to manipulate angles in this way is an essential skill in algebraic problem solving, particularly in trigonometry and calculus.