Question

The value of $\cos^{-1}\left(\cos\left(\frac{7\pi}{6}\right)\right)$ is

• A. $\frac{5\pi}{6}$
• B. $\frac{\pi}{3}$
• C. $\frac{7\pi}{6}$
• D. $\frac{\pi}{6}$

Hint:

Always check the principal value range of inverse trigonometric functions before writing the final answer.

Answer 1

The Correct Option is A

Step 1: Understanding the principal value range.
The principal value range of $\cos^{-1}x$ is $[0,\pi]$. Hence, the answer must lie within this interval.
Step 2: Evaluating the cosine value.
\[ \cos\left(\frac{7\pi}{6}\right) = \cos\left(\pi + \frac{\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \]
Step 3: Finding the angle in the principal range.
The angle in $[0,\pi]$ whose cosine is $-\frac{\sqrt{3}}{2}$ is \[ \frac{5\pi}{6} \]
Step 4: Conclusion.
\[ \cos^{-1}\left(\cos\left(\frac{7\pi}{6}\right)\right) = \frac{5\pi}{6} \]