Problem 65 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$\frac{23 \pi}{5}$$

Step-by-Step Solution

Answer

The positive angle less than \(2 \pi\) that is coterminal with \(\frac{23 \pi}{5}\) is \(\frac{3 \pi}{5}\).

Step 1: Determine the number of complete revolutions

Determine how many complete \(2 \pi\) revolutions the given angle \(\frac{23 \pi}{5}\) contains. To do this, divide \(\frac{23 \pi}{5}\) by \(2 \pi\). That is done as follows: \(\frac{23}{5} ÷ 2\). A complete revolution is any whole number quotient from this division. The result of the division is \(2\) remainder \( \frac{3}{5}\). Meaning, the angle \(\frac{23 \pi}{5}\) is equivalent to \(2\) complete revolutions and an additional \(\frac{3 \pi}{5}\).

Step 2 Setting the coterminal angle

Having completed \(2\) full revolutions, the coterminal angle to \(\frac{23 \pi}{5}\) is the angle equal to the remaining portion of \(2\pi\). This is \(\frac{3 \pi}{5}\). Thus, the angle equal to \(\frac{3 \pi}{5}\) is the unique positive angle less than \(2 \pi\) that is coterminal to \(\frac{23 \pi}{5}\).