
Find the principal value of $\cos e{{c}^{-1}}\left( 2 \right)$.
Answer
474.6k+ views
Hint: We will be using the concept of inverse trigonometric functions to solve the problem. We will first write 2 as $\cos ec\theta $ then we will use the fact that $\cos e{{c}^{-1}}\left( \cos ecx \right)=x$ for $x\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]-\left\{ 0 \right\}$.
Complete step by step answer:
Now, we have to find the value of $\cos e{{c}^{-1}}\left( 2 \right)$.
Now, we will first represent 2 in terms of cosecant of an angle. So, we know that the value of $\cos ec\left(\dfrac{\pi }{6} \right)$ is 2.
$2=\cos ec\left( \dfrac{\pi }{6} \right).........\left( 1 \right)$
We have taken $2=\cos ec\left(\dfrac{\pi }{6}\right)$ as in the view of the principal value convention x is confined to$x\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]-\left\{ 0 \right\}$.
Now, we know that the graph of $\cos e{{c}^{-1}}\left( \cos ecx \right)$ is,
Now, we have to find the value of$\cos e{{c}^{-1}}\left( 2 \right)$.
We will use the equation (1) to substitute the value of 2. So, we have,
$\cos e{{c}^{-1}}\left( \cos ec\left( \dfrac{\pi }{6} \right) \right)$
Also, we know that $\cos e{{c}^{-1}}\left( \cos ecx \right)=x$. So, we have,
$\cos e{{c}^{-1}}\left( \cos ec\left( \dfrac{\pi }{6} \right) \right)=\dfrac{\pi }{6}$
So, the correct answer is “$\dfrac{\pi }{6}$”.
Note: To solve these type of question it is important to note that we have used a fact that $\cos e{{c}^{-1}}\left( \cos ecx \right)=x$ only for $x\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]-\left\{ 0 \right\}$. For another value of x the graph of $\cos e{{c}^{-1}}\left( \cos ecx \right)$ must be used to find the value.
Complete step by step answer:
Now, we have to find the value of $\cos e{{c}^{-1}}\left( 2 \right)$.
Now, we will first represent 2 in terms of cosecant of an angle. So, we know that the value of $\cos ec\left(\dfrac{\pi }{6} \right)$ is 2.
$2=\cos ec\left( \dfrac{\pi }{6} \right).........\left( 1 \right)$
We have taken $2=\cos ec\left(\dfrac{\pi }{6}\right)$ as in the view of the principal value convention x is confined to$x\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]-\left\{ 0 \right\}$.
Now, we know that the graph of $\cos e{{c}^{-1}}\left( \cos ecx \right)$ is,

Now, we have to find the value of$\cos e{{c}^{-1}}\left( 2 \right)$.
We will use the equation (1) to substitute the value of 2. So, we have,
$\cos e{{c}^{-1}}\left( \cos ec\left( \dfrac{\pi }{6} \right) \right)$
Also, we know that $\cos e{{c}^{-1}}\left( \cos ecx \right)=x$. So, we have,
$\cos e{{c}^{-1}}\left( \cos ec\left( \dfrac{\pi }{6} \right) \right)=\dfrac{\pi }{6}$
So, the correct answer is “$\dfrac{\pi }{6}$”.
Note: To solve these type of question it is important to note that we have used a fact that $\cos e{{c}^{-1}}\left( \cos ecx \right)=x$ only for $x\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]-\left\{ 0 \right\}$. For another value of x the graph of $\cos e{{c}^{-1}}\left( \cos ecx \right)$ must be used to find the value.
Recently Updated Pages
Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Master Class 12 Physics: Engaging Questions & Answers for Success

Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Trending doubts
The probability that a leap year will have only 52 class 12 maths CBSE

The final image formed by a compound microscope is class 12 physics CBSE

Which of the following properties of a proton can change class 12 physics CBSE

Sum of two skew symmetric matrices is always matri class 12 maths CBSE

What is the difference between at least one head and class 12 maths CBSE

What is the energy band gap of silicon and germanium class 12 physics CBSE
