# Find the value of

Question:

Find the value of $\cos ^{-1}\left(\cos \frac{15 \pi}{6}\right)$

Solution:

We know that $\cos ^{-1}(\cos x)=x$ if $x \in[0, \pi]$, which is the principal value branch of $\cos ^{-1} x$.

Here, $\frac{13 \pi}{6} \notin[0, \pi]$.

Now, $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)$ can be written as:

$\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(2 \pi+\frac{\pi}{6}\right)\right]=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]$, where $\frac{\pi}{6} \in[0, \pi]$

$\therefore \cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]=\frac{\pi}{6}$
The text passage contains a mathematical question and provides a detailed solution to it.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Find the value of

Question:

Find the value of $\cos ^{-1}\left(\cos \frac{15 \pi}{6}\right)$

Solution:

We know that $\cos ^{-1}(\cos x)=x$ if $x \in[0, \pi]$, which is the principal value branch of $\cos ^{-1} x$.

Here, $\frac{13 \pi}{6} \notin[0, \pi]$.

Now, $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)$ can be written as:

$\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(2 \pi+\frac{\pi}{6}\right)\right]=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]$, where $\frac{\pi}{6} \in[0, \pi]$

$\therefore \cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(\frac{\pi}{6}\right)\right]=\frac{\pi}{6}$
user
The text passage contains a mathematical question and provides a detailed solution to it.

TYPE: SINGLE_QUESTION_WITH_ANSWER