Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Find the angle of least positive measure that is co terminal with the given angle. $$-40^{\circ}$$

Step-by-Step Solution

Answer

The angle of least positive measure that is coterminal with \(-40^{\circ}\) is \(320^{\circ}\).

Step 1: Definition of Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. An angle of least positive measure that is coterminal with a given angle can be found by adjusting the given angle until it falls within the standard position of 0° to 360°.

Step 2: Adding 360 Degrees to the Angle

To find a coterminal angle that is positive, add 360° to the given angle of \(-40^{\circ}\). This is because 360° represents a full circle, and adding it does not change the direction of the angle. Calculate: \(-40^{\circ} + 360^{\circ} = 320^{\circ}\).

Step 3: Conclusion

Since \(320^{\circ}\) falls within the range of 0° to 360°, it is a positive angle that is coterminal with \(-40^{\circ}\). Hence, the angle of least positive measure that is coterminal with \(-40^{\circ}\) is \(320^{\circ}\).

Key Concepts

Angle Measurement

Understanding angle measurement is crucial for finding coterminal angles. Angles can be measured in degrees, which signify the amount of rotation from the initial side to the terminal side.
To visualize this, imagine placing a point on a circle's center and drawing a line to the edge, which is your initial side. As you rotate the line around the center, it creates an angle with the original position.
There are 360 degrees in a full circle, meaning if you continue rotating from the initial position, at 360°, you will end up at the same position where you started.
It's essential to note that angles are not limited to just one complete revolution. They can exceed 360° or be negative, indicating a direction opposite to the positive or counter-clockwise direction. However, negative angles, like -40°, often need to be converted for easier interpretation in problems, which involves understanding coterminal concepts.
Recognizing these measurements ensures clarity when working with angles in various mathematical tasks.

Positive Angles

A positive angle measures the rotation in a counter-clockwise direction from the initial side to the terminal side of an angle. Starting from the initial side, the rotation moves counter-clockwise, resulting in a positive value.
In cases where an angle is given as negative, such as -40°, finding a coterminal positive angle helps align it within the standard range of 0° to 360°. This makes calculations and interpretations more consistent.
By converting a negative angle to a positive one, you essentially "unwrap" the rotation in the negative direction by adding a full circle (360°), establishing an equivalent angle that fits into the positive framework.
For example, adding 360° to -40° results in a positive angle of 320°, which is now easy to graph and understand as it represents the same spatial orientation as -40°, minus the negative complexity.

360 Degrees

The significance of 360 degrees lies in its representation of a complete rotation or full circle. Every angle measured in degrees relies on this full-circle concept.
When dealing with coterminal angles, adding 360° to any angle does not change its direction or orientation, but simply lets you find an equivalent angle within the desired range.
For example, take -40°, a negative angle that needs to be made positive for conventional purposes. By adding 360°, you arrive at 320°, an angle that falls cleanly within the 0° to 360° range.
Understanding that 360° doesn't alter the angle’s direction allows easy conversion to a standard angle measure while knowing that the direction and position in the circle remain unchanged. This helps maintain consistency in representing all angles irrespective of their initial negative or greater-than-360-degree position.