Problem 66 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$\frac{25 \pi}{6}$$

Step-by-Step Solution

Answer

The positive coterminal angle less than \(360^{\circ}\) or \(2 \pi\) with the given angle \(\frac{25 \pi}{6}\) is \( \frac{\pi}{6} \).

Step 1: Convert the angle to less than \(2\pi\)

Since \( 2 \pi \) corresponds to one complete turn around the circle, we can subtract multiples of \(2\pi\) from \(\frac{25\pi}{6}\) to get it in the interval from \(0\) to \(2\pi\). The general formula to convert an angle to an equivalent one less than \(2\pi\) is \( \theta - 2\pi n \) where \(n\) is a positive integer, chosen so that \(0\leq \theta - 2\pi n< 2\pi\). In this case, \( n \) is determined by the relation \( n = \lfloor \frac{25}{6} \rfloor \) where \(\lfloor x \rfloor\) is the greatest integer less than or equal to \(x\).

Step 2: Find the equivalent coterminal angle

Substitute \(n = 4\) (since the integer part of \( \frac{25}{6} \) is 4) into the equation from the previous step, we get \( \frac{25\pi}{6} - 2\pi \times 4 \), which simplifies to \( \frac{25\pi}{6} - \frac{24\pi}{6} = \frac{\pi}{6} \). Thus, the angle that's coterminal to \( \frac{25\pi}{6} \) and within the interval from \(0\) to \(2\pi\) is \( \frac{\pi}{6} \).