Problem 69 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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: \(-\frac{31\pi}{7} + 3(2\pi) = -\frac{31\pi}{7} + \frac{42\pi}{7} = \frac{11\pi}{7}\)

Step 3: 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.