Problem 70 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$-\frac{38 \pi}{9}$$

Step-by-Step Solution

Answer

The positive angle less than \(2\pi\), which is coterminal with \(-\frac{38\pi}{9}\), is \(\frac{16\pi}{9}\).

Step 1: Identify Coterminal Angles

Coterminal angles are angles that share the same terminal side. They could be found by adding or subtracting multiples of \(2\pi\) or \(360^{\circ}\) (a full revolution) to the given angle. So, we need to add multiples of \(2\pi\) to \(-\frac{38 \pi}{9}\) until we get a positive angle less than \(2\pi\).

Step 2: Add Multiples of \(2\pi\)

Finding the proper value to add can be difficult so let's try to simplify the task. We know that \(2\pi \times 1 = 2\pi\) or \(\frac{18 \pi}{9}\), \(2\pi \times 2 = 4\pi\) or \(\frac{36 \pi}{9}\), \(2\pi \times 3 = 6\pi\) or \(\frac{54 \pi}{9}\) and so on until we find a positive angle less than \(2\pi\). Try to aim for a multiple that will result in an angle slightly larger than \(\frac{38\pi}{9}\) but remember the result should be less than \(2\pi\) or \(\frac{18 \pi}{9}\). This may take several trials and errors.

Step 3: Compute the Positive Coterminal Angle

For example, it was found that 3 rotations or \(6\pi\) or \(\frac{54 \pi}{9}\) added to \(-\frac{38\pi}{9}\) gives a positive coterminal angle. When added together, we get \(-\frac{38\pi}{9} + \frac{54\pi}{9} = \frac{16\pi}{9}\). This is our positive coterminal angle that is less than \(2\pi\).