Problem 60 - Algebra and Trigonometry

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$ -160^{\circ} $$

Step-by-Step Solution

Answer

A positive angle that is coterminal with -160 degrees is 200 degrees.

Step 1: Understanding Coterminal Angles

Two angles are coterminal if they share the same terminal side. This means that we can add or subtract multiples of 360 degrees (or \(2 \pi\) in radian measurement) from any angle, and the result will be an angle that is coterminal with the original.

Step 2: Getting equivalent positive angle

Find an equivalent positive angle to -160 degrees by adding 360 degrees because the question specifies a positive angle. By adding 360 degrees to -160 degrees, it gives \(360^{\circ} - 160^{\circ} = 200^{\circ}\). Hence, 200 degrees is a positive angle coterminal with -160 degrees.

Step 3: Verifying the angle

The last step is to verify that the obtained angle indeed shares a terminal side with -160 degrees, and that it is less than 360 degrees. The angle of 200 degrees fulfills both these conditions, and hence is a valid solution.

Key Concepts

Angle Measurement

Understanding angles and how they are measured is essential in trigonometry and geometry. Angles are formed by two rays originating from a common endpoint, known as the vertex. The size of an angle is determined by the amount of rotation from one ray to the other. To measure angles, we use units such as degrees or radians.

Degrees are the most common unit of angle measurement. A complete revolution is divided into 360 equal parts, and each part is one degree, denoted as \(^{\circ}\). Therefore, a right angle, which is one-quarter of a full revolution, measures 90 degrees.

When working with angles, it's crucial to understand their positive and negative orientations. Positive angles result from a counterclockwise rotation, whereas negative angles stem from a clockwise rotation. This concept is pivotal when finding coterminal angles, as it allows us to adjust a given angle by adding or subtracting full rotations.

Radian Conversion

Radian is another unit of angle measurement, often used in higher mathematics like calculus. While degrees measure an angle based on dividing a circle into 360 parts, radians do so based on the circle's radius and arc length. One radian is the angle created when the arc length equals the radius of the circle.

A complete circle measures \(2\pi\) radians, because the circumference of a circle is \(2\pi\) times the radius. A useful relationship to remember is that \(360^{\circ} = 2\pi\) radians, helping us convert between these units.

To convert degrees to radians, multiply the angle in degrees by \(\frac{\pi}{180}\). Conversely, to transfer radians to degrees, multiply the radians by \(\frac{180}{\pi}\). This conversion is useful in determining coterminal angles in various units.

Coterminality Concept

Coterminality is the concept where two angles share the same terminal side after a complete revolution. This occurs because adding or subtracting a multiple of a full circle results in the same angle position.

To find coterminal angles in degrees, you can add or subtract multiples of 360 degrees since one full circle equals \(360^{\circ}\). For example, to find the coterminal angle for \(-160^{\circ}\), add \(360^{\circ}\) to obtain \(200^{\circ}\), which is the equivalent positive angle.

Similarly, in radians, you can add or subtract multiples of \(2\pi\). This technique ensures that while the numerical value of the angle may change, its position relative to the circle remains unchanged. Understanding coterminality helps in simplifying complex angle problems and verifying solutions.