Which angle is adjacent to $\frac{\pi }{4}$ on the unit circle?

• A. $\frac{\pi }{8}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{5}$

Answer: (A)

Verified step by step guidance

1. Recall that on the unit circle, two angles are adjacent if they differ by the smallest positive angle between standard positions, often corresponding to the difference between consecutive points on the circle.
2. Identify the given angle, which is \(\frac{\pi}{4}\) radians, and consider the angles provided as options: \(\frac{\pi}{8}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), and \(\frac{\pi}{5}\).
3. Calculate the absolute difference between \(\frac{\pi}{4}\) and each option to find which one is closest to \(\frac{\pi}{4}\), indicating adjacency on the unit circle.
4. Compare the differences: for example, \(\left| \frac{\pi}{4} - \frac{\pi}{8} \right| = \frac{\pi}{8}\), \(\left| \frac{\pi}{4} - \frac{\pi}{3} \right| = \frac{\pi}{12}\), and so on.
5. Determine which difference corresponds to the smallest positive angle, and that angle will be adjacent to \(\frac{\pi}{4}\) on the unit circle.