Problem 63 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$\frac{19 \pi}{6}$$

Step-by-Step Solution

Answer

The positive coterminal angle less than \(2 \pi\) that is equivalent to \(\frac{19 \pi}{6}\) is \(\frac{7 \pi}{6}\).

Step 1: Understand How Coterminal Angles are Calculated

Two angles are coterminal if they differ by an integral multiple of \(2\pi\) or \(360^{\circ}\). You can determine a positive coterminal angle that is less than \(2\pi\) by subtracting multiples of \(2\pi\) from the original angle \(\frac{19 \pi}{6}\) until the angle is less than \(2\pi\).

Step 2: Determine The Multiples of \(2\pi\)

Subtract multiples of \(2\pi\) from the original angle to get an equivalent angle between \(0\) and \(2\pi\). A thing to note is that \(2\pi\) is the same as \(\frac{12\pi}{6}\) to have the same denominators: \(\frac{19 \pi}{6} - n(\frac{12 \pi}{6})\).

Step 3: Calculate the Coterminal Angle

To calculate the coterminal angle, you can subtract \(2 \pi\) (i.e., \(\frac{12 \pi}{6}\)) twice from \(\frac{19 \pi}{6}\) to get: \(\frac{19 \pi}{6} - 2(\frac{12 \pi}{6}) = \frac{19 \pi}{6} - \frac{24 \pi}{6} = \frac{-5 \pi}{6}\). However, this is a negative coterminal angle, so add 2\pi (or \(\frac{12 \pi}{6}\)) to get a positive counterpart: \(\frac{-5 \pi}{6} + \frac{12 \pi}{6} = \frac{7\pi}{6}\).