Problem 72 - Precalculus

Question

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$-187^{\circ}$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle is \(173^{\circ}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find a positive coterminal angle, we need to add or subtract full circles (360°) until we get a positive angle.

Step 2: Adding 360° to Find a Positive Angle

Since we start with a negative angle, we add 360° to \[-187^{\circ}\] to find a coterminal angle:\[-187^{\circ} + 360^{\circ} = 173^{\circ}.\]

Step 3: Verify the Positive Angle

Check if the angle \(173^{\circ}\) is positive and less than 360°. Since it is, \(173^{\circ}\) is indeed the smallest positive coterminal angle.

Key Concepts

Understanding Angle Measurement

Angle measurement is a way to quantify the amount of rotation between two intersecting lines having a common point called the vertex. This measurement can be in degrees or radians. For the current discussion, let's focus on degree measurement as it is one of the most common methods. To visualize, think of a circular protractor divided into 360 equal parts---each part is one degree. Hence, a complete revolution in a circle is 360°. This divides the angle measurement into:

• Acute Angles: Less than 90°
• Right Angles: Exactly 90°
• Obtuse Angles: Between 90° and 180°
• Straight Angles: Exactly 180°

Understanding these categories can help you easily identify coterminal angles when given a problem.

Exploring Positive Angles

Positive angles are those angles that are measured in the counterclockwise direction from the initial side. In the context of coterminal angles, when you are tasked with finding a positive angle that is coterminal with a given angle, you may need to adjust the original measurement.For instance, if you have a negative angle like \[-187^{\circ}\], this signifies a clockwise rotation. To convert this into a positive angle, you add a full circle, or 360°. By doing so, \[-187^{\circ} + 360^{\circ} = 173^{\circ}\], getting a positive angle. This simple adjustment is crucial since coterminal angles can be both positive and negative, but the smallest positive measure is often the sought-after solution in these exercises.

Concept of Degree Measure

The degree measure is a unit of angle measurement which divides one complete rotation into 360 equal parts. This method is deeply integrated into various mathematical contexts and practical applications around the world.Degree measure plays a critical role in understanding coterminal angles. To determine if two angles, like \(360n + \theta\), where \(n\) is an integer, are coterminal, you can compare their degree measures. When working with exercises involving negative angles, converting them, as shown in previous steps, ensures one finds the smallest positive coterminal angle.By consistently applying the knowledge that 360° represents one full revolution, solving such problems becomes intuitive. This approach is valuable because it helps further explore the vast applications of angle relationships in geometry.