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Chapter 4: Problem 16

Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians. $$\text { (a) } \frac{2 \pi}{3} \quad \text { (b) }-\frac{9 \pi}{4}$$

Short Answer

Expert verified
The two coterminal angles for \( \frac{2\pi}{3} \) are \( \frac{8\pi}{3} \) and \( -\frac{4\pi}{3} \). The two coterminal angles for \( -\frac{9\pi}{4} \) are \( \frac{7\pi}{4} \) and \( -\frac{17\pi}{4} \).

Step by step solution

01

Determine the positive coterminal for \( \frac{2\pi}{3} \)

To compute the positive coterminal angle, add \(2\pi\) to the given angle. So, \( \frac{2\pi}{3} + 2\pi = \frac{2\pi}{3} + \frac{6\pi}{3} = \frac{8\pi}{3} \).
02

Determine the negative coterminal for \( \frac{2\pi}{3} \)

To find the negative coterminal angle, subtract \(2\pi\) from the original angle. So, \( \frac{2\pi}{3} - 2\pi = \frac{2\pi}{3} - \frac{6\pi}{3} = -\frac{4\pi}{3} \).
03

Determine the positive coterminal for \( -\frac{9\pi}{4} \)

To find the positive coterminal angle, add \(2\pi\) to the original angle. So, \( -\frac{9\pi}{4} + 2\pi = -\frac{9\pi}{4} + \frac{8\pi}{4} = -\frac{1\pi}{4} \). Note that we add \(2\pi\) until the result is positive. Therefore, the actual positive coterminal is \( -\frac{1\pi}{4} + 2\pi = 7\pi/4 \).
04

Determine the negative coterminal for \( -\frac{9\pi}{4} \)

We find the negative coterminal angle by subtracting \(2\pi\) from the original angle until the result is negative. So, \( -\frac{9\pi}{4} - 2\pi = -\frac{9\pi}{4} - \frac{8\pi}{4} = -\frac{17\pi}{4} \).

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