Problem 29 - Precalculus: A Concise Course

Question

Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians. (a) \(\theta=\frac{2 \pi}{3}\) (b) \(\theta=\frac{\pi}{12}\)

Step-by-Step Solution

Answer

The coterminal angles for (a) \(\theta=\frac{2\pi}{3}\) are \(\frac{8\pi}{3}\) (positive) and \(\frac{-4\pi}{3}\) (negative), and for (b) \(\theta=\frac{\pi}{12}\) are \(\frac{25\pi}{12}\) (positive) and \(\frac{-23\pi}{12}\) (negative).

Step 1: Identifying Original Angles

(a) The original angle in radians is \(\frac{2\pi}{3}\) and (b) the original angle in radians is \(\frac{\pi}{12}\).

Step 2: Determining Coterminal Angles for \(\theta=\frac{2\pi}{3}\)

To find coterminal angles for \(\frac{2\pi}{3}\), add and subtract \(2\pi\) to/from it. The positive coterminal angle is \(\frac{2\pi}{3}+2\pi=\frac{8\pi}{3}\). The negative coterminal angle is \(\frac{2\pi}{3}-2\pi=\frac{-4\pi}{3}\).

Step 3: Determining Coterminal Angles for \(\theta=\frac{\pi}{12}\)

To find coterminal angles for \(\frac{\pi}{12}\), add and subtract \(2\pi\) to/from it. The positive coterminal angle is \(\frac{\pi}{12}+2\pi=\frac{25\pi}{12}\). The negative coterminal angle is \(\frac{\pi}{12}-2\pi=\frac{-23\pi}{12}\).