Problem 42 - Algebra and Trigonometry

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ -100^{\circ} $$

Step-by-Step Solution

Answer

The angle coterminal with \(-100^{\circ}\) is \(260^{\circ}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side after a full rotation. To find a coterminal angle between \(0^{\circ}\) and \(360^{\circ}\), we can add or subtract multiples of \(360^{\circ}\) from the given angle until it falls within this range.

Step 2: Initial Adjustment

Start with the given angle \(-100^{\circ}\). Since it is negative, we need to add \(360^{\circ}\) to bring it into the positive range. Calculate: \(-100^{\circ} + 360^{\circ} = 260^{\circ}\).

Step 3: Verification of the Range

Check if the resulting angle \(260^{\circ}\) is between \(0^{\circ}\) and \(360^{\circ}\). Since it falls within this range, \(260^{\circ}\) is the desired coterminal angle.

Key Concepts

Angle Measurement

Understanding angle measurement is a key concept when studying coterminal angles. Angles are most commonly measured in degrees, which represent how far around a circle something is turned. A full circle has 360 degrees. The direction in which you measure an angle can influence its value and properties. There are different units of angle measurement, such as:

• Degrees, which divide a circle into 360 equal parts.
• Radians, which are based on the radius of the circle and are another standard unit of measurement.

When dealing with coterminal angles, it’s important to understand that these are angles that share the same finishing point or "terminal side" when plotted on a circular grid, despite possibly having different measurements due to full rotations. The process of finding those involves adding or subtracting full circle measures (360 degrees) until reaching the desired range.

Positive Angle

A positive angle results when an angle is measured in the counterclockwise direction from its initial side on the x-axis. This direction is typically used to measure regular angles within mathematical concepts. When working with angles like -100^ in the exercise, you need to adjust them to become positive. This involves adding a full revolution (360 degrees) to the negative angle until the angle falls within the range of 0 to 360 degrees. For a negative angle, such as -100 degrees, adding 360 degrees results in a positive angle of 260 degrees. These positive angles are essential for accurately describing coterminal angles within the specific range, and making use of positive angles ensures clarity, especially when these measurements are applied in various practical contexts.

Angle Rotation

Angle rotation is a fundamental concept that involves turning around a central point. Understanding how rotation affects angle measurement is crucial when determining coterminal angles. Here are some basic ideas about angle rotation:

• Counterclockwise rotation: Normally results in positive angles.
• Clockwise rotation: Typically gives negative angles.

Coterminal angles emerge from the property of rotation because you can start at the same initial side and rotate multiple times around a circle. The angles will still look the same on the graph. This trick is handy when trying to find coterminal angles within a certain range. For example, you took -100 degrees and added 360 degrees to execute a full rotation, landing on 260 degrees, which aligns perfectly as a coterminal angle within the standard 0 to 360-degree range.