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Chapter 4: Problem 122

Determine whether the statement is true or false. Justify your answer. The difference between the measures of two coterminal angles is always a multiple of \(360^{\circ}\) if expressed in degrees and is always a multiple of \(2 \pi\) radians if expressed in radians.

Short Answer

Expert verified
The statement is true. The difference between the measures of two coterminal angles is always a multiple of \(360^{\circ}\) if expressed in degrees and is always a multiple of \(2 \pi\) radians if expressed in radians. The justification is the definition of coterminal angles itself, which have a common terminal side, indicating they differ by a certain number of complete revolutions or cycles.

Step by step solution

01

Define Coterminal Angles

Coterminal angles are angles in a standard position (angles with the initial side on the positive x-axis) that have a common terminal side. For example, angles of 30掳, 390掳 and -330掳 are all coterminal.
02

Understand the difference between coterminal angles

The difference between the measures of two coterminal angles is a multiple of a complete revolution. In degree measure, a full revolution is \(360^{\circ}\) and in radian measure, it is \(2 \pi\) radians. This is a property of coterminal angles.
03

Verify the statement with examples

Take an example where the first angle is 30掳 and its coterminal angle is 390掳. The difference is 390掳 - 30掳 = 360掳, which is a multiple of \(360^{\circ}\). In radians, let's consider an angle of \(\pi / 6\) radians and its coterminal angle of \(13\pi / 6\). The difference is \(13\pi / 6 - \pi / 6 = 2 \pi\), which is indeed a multiple of \(2 \pi\) radians.

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