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Chapter 4: Problem 69

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$-\frac{31 \pi}{7}$$

Short Answer

Expert verified
The positive angle less than \(2\pi\) radians and coterminal with \(-\frac{31\pi}{7}\) is \(\frac{11\pi}{7}\) radians.

Step by step solution

01

Calculate the number of full rotations

Calculate the number of full rotations in \(-\frac{31\pi}{7}\) since \(2\pi\) represents a full rotation. This is done by dividing \(-\frac{31\pi}{7}\) by \(2\pi\) or simply dividing \(-\frac{31}{7}\) by 2 which gives approximately -2.2. Since we need an integer number of rotations, round down to -3 because our original angle is negative.
02

Add the full rotations to the original angle

Since each full rotation is \(2\pi\) radians, add three full rotations to the original angle. This is calculcated as follows: \(-\frac{31\pi}{7} + 3(2\pi) = -\frac{31\pi}{7} + \frac{42\pi}{7} = \frac{11\pi}{7}\)
03

Verify the result

Finally, verify that the resulting angle \(\frac{11\pi}{7}\) is positive and less than \(2\pi\) radians. Since \(2 < \frac{11}{7} < 3\), it confirms that the obtained radian measure lies between 0 and \(2\pi\) radians as required.

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