Problem 59 - Algebra and Trigonometry

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$ -150^{\circ} $$

Step-by-Step Solution

Answer

The positive coterminal angle to \(-150^{\circ}\) is \(210^{\circ}\).

Step 1: Understanding the problem

Start by understanding the concept of coterminal angles. Two angles are coterminal if they have the same terminal side. This means that you can add or subtract multiples of 360 degrees or \(2 \pi\) radian to an angle, and the resulting angle will be coterminal to the original one.

Step 2: Find a coterminal angle in the positive direction

The angle is given as \(-150^{\circ}\). One way to find a positive coterminal angle is to keep adding 360 degrees until you get a positive result. So let's add 360 degrees to \(-150^{\circ}\): \(-150^{\circ} + 360^{\circ} = 210^{\circ}\). This is a value greater than 0 and less than 360, so it is a positive coterminal angle.

Step 3: Check the result

In this case, \(210^{\circ}\) is indeed a positive angle less than \(360^{\circ}\) that is coterminal with \(-150^{\circ}\). So, the positive coterminal angle is \(210^{\circ}\).

Key Concepts

Angle Measurement

Angles are a fundamental concept in geometry and trigonometry that describe a measure of rotation. They tell us how much a turning body, or a ray, deviates from a set position, usually the baseline.

• Measured in degrees or radians.
• Key for navigating directions, architecture, astronomy, and many fields of science and engineering.

In degrees, a circle is divided into 360 equal parts. Thus, a full rotation around a circle measures 360 degrees. Using the radian measure, angles are described using the radius of a circle. For example, a full circle is 2π radians. This is because the circumference of the circle, in terms of the radius, is 2π times the radius itself. Understanding that angles can be represented in both degrees and radians allows us to switch perspectives and make calculations across different scientific fields. It also enables better grasping of concepts like coterminal angles, multipliers (like 360 or 2π), and conversions between units.

Positive Angles

When we talk about positive angles, we refer to angles measured in the counterclockwise direction from the initial side, typically a ray on the positive x-axis.

• Measured counterclockwise.
• Value is greater than 0 degrees/radians and less than 360 degrees or 2π radians.

In the context of coterminal angles, it's important because we often look for the smallest positive representation of an angle. For instance, if we have an angle like -150°, adding 360° provides us with 210°, a positive angle. This is crucial as certain applications, like navigating, often require positive measurements. Always remember: the goal is to locate the positive angle that is less than one full rotation, represented by either 360 degrees or 2π radians.

Degrees and Radians

Degrees and radians are two systems to measure the same thing: angles. Both are useful, and each has its context where it shines best.

• Degrees: More intuitive and often easier to visualize. Used frequently in everyday life and navigation.
• Radians: Based on the radius of a circle and crucial for mathematical calculations and theories.

Radians have a direct correlation to circular motion, since they relate the radius of the circle to its circumference. Hence 2π radians equals one full circle. When looking at coterminal angles, understanding the conversion between degrees and radians is essential, especially in physics and calculus, where radians are often used. To convert:

• From degrees to radians: Multiply the degree measure by (π/180).
• From radians to degrees: Multiply the radian measure by (180/π).

Knowing these relationships is key to gaining flexibility and understanding in trigonometry and forms the basis for solving many practical problems.