Problem 64 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$\frac{17 \pi}{5}$$

Step-by-Step Solution

Answer

The positive angle less than \(2\pi\) which is coterminal with \(\frac{17\pi}{5}\) is \(\frac{2\pi}{5}\) radians.

Step 1: Given angle in radians

We are given the angle \( \frac{17\pi}{5}\) in radians.

Step 2: Identifying the coterminal angle

We understand that angles are coterminal if they end up at the same position on the circle when starting from the origin. We can achieve this by adding or subtracting multiples of complete rotations, which is \(2\pi\) radians or \(360^\circ\). So, to find a positive coterminal angle less than \(2\pi\), subtract multiples of \(2\pi\) until the resultant angle falls within the range of 0 and \(2\pi\).

Step 3: Subtract multiples of \(2 \pi\)

Subtract multiples of \(2\pi\) from the given angle \(\frac{17\pi}{5}\) until we get a value less than \(2\pi\). The largest multiple of \(2\pi\) that is less than \(\frac{17\pi}{5}\) is \(2*3\pi = 6\pi\). Hence, perform the subtraction: \(\frac{17\pi}{5} - 6\pi = \frac{2\pi}{5}\).

Step 4: The coterminal angle is

So the positive angle less than \(2\pi\) that is coterminal with \(\frac{17\pi}{5}\) is \(\frac{2\pi}{5}\) radians.