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Chapter 1: Problem 22

Determine whether the angles in each given pair are coterminal. $$ 20^{\circ}, 380^{\circ} $$

Short Answer

Expert verified
Yes, 20掳 and 380掳 are coterminal angles.

Step by step solution

01

- Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but may have different measures. They can be found by adding or subtracting full rotations (360掳) to or from the given angles.
02

- Find Coterminal Angle for 20掳

To check if 380掳 is a coterminal angle to 20掳, subtract 360掳 from 380掳: \(380^{\text{掳}} - 360^{\text{掳}} = 20^{\text{掳}}\)
03

- Compare the Angles

Since the result of the subtraction is 20掳, which matches the given first angle, 20掳 and 380掳 are coterminal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Measurement
Angles are measured in degrees using a protractor. One full rotation around a point is 360掳, which is known as a complete circle.
Sometimes, angles can go beyond 360掳, resulting in measurements greater than one full rotation.
For example, 380掳 means we have gone a full circle (360掳) and then an extra 20掳.
Knowing how to measure and interpret angles is crucial for understanding coterminal angles.
Coterminal Angles
Coterminal angles are angles that end up at the same position when drawn in a circular path.
They have different measurements but share the same initial and terminal sides. You can find coterminal angles by adding or subtracting multiples of 360掳 to the given angle.
For instance, the angles 20掳 and 380掳 from our exercise are coterminal because 380掳 is simply 20掳 plus a full turn of 360掳: 380掳 = 20掳 + 360掳.
This property of angles helps in many areas like circular motion and trigonometrical calculations.
Angle Subtraction
To determine if two angles are coterminal, you can subtract one angle from the other and see if the result is a multiple of 360掳.
Let's take 380掳 and subtract 360掳: 380掳 - 360掳 = 20掳.
Because the result is 20掳, which is the same as our original angle, we confirm that 20掳 and 380掳 are coterminal.
This subtraction method is handy for quickly identifying whether two angles are coterminal.
Circular Motion
Circular motion involves objects moving in a circular path.
The concept of coterminal angles applies heavily here since the position of the object in circular motion often relies on angles.
For example, if an object starts at 0掳 and moves to 720掳 in two full rotations, it is coterminal with the starting 0掳.
Understanding coterminal angles helps in predicting and analyzing movements in systems involving circular paths, such as gears and wheels.

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