Answer

Verified

449.1k+ views

Hint: The question is related to inverse trigonometric functions. Assume the given function to be equal to $x$. Find the value of $\cot x$. Then find the value of $x$ which gives the acquired value on applying cotangent function.

Complete step-by-step answer:

We are asked to find the principal value of the inverse trigonometric function ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$. Let us assume the value of the inverse trigonometric function to be equal to $x$. So, we get:

${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=x$

Now, we will apply cotangent function on both sides of the equation. On applying cotangent function on both sides of the equation, we get:

\[\cot \left( {{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right) \right)=\cot x\]

Now, we know the value of $\cot \left( {{\cot }^{-1}}y \right)$ is equal to $y$. So, we get:

$\tan \dfrac{3\pi }{4}=\cot x.....(i)$.

Now, we know, tangent function is negative in the second quadrant. So, the value of $\tan \dfrac{3\pi }{4}$ is equal to $-1$ . We will substitute the value of $\tan \dfrac{3\pi }{4}$ as $-1$ in equation $(i)$. On substituting the value of $\tan \dfrac{3\pi }{4}$ as $-1$ in equation $(i)$, we get:

$\cot x=-1$.

We know, the range for principal value is $\left( 0,\pi \right)$. So, we have to find a value of $x$ such that $x\in \left( 0,\pi \right)$ and $\cot x=-1$. The only possible value which satisfies both conditions is $x=\dfrac{3\pi }{4}$.

So , the value of principal value of the inverse trigonometric function ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$ is equal to $\dfrac{3\pi }{4}$.

Note: While solving the problem, make sure that the value of the inverse trigonometric function lies in the principal value range, i.e. $\left( 0,\pi \right)$for \[cot\] function. Students generally forget this condition and end up getting a wrong answer. So, this condition must be satisfied by the obtained principal value.

Complete step-by-step answer:

We are asked to find the principal value of the inverse trigonometric function ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$. Let us assume the value of the inverse trigonometric function to be equal to $x$. So, we get:

${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)=x$

Now, we will apply cotangent function on both sides of the equation. On applying cotangent function on both sides of the equation, we get:

\[\cot \left( {{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right) \right)=\cot x\]

Now, we know the value of $\cot \left( {{\cot }^{-1}}y \right)$ is equal to $y$. So, we get:

$\tan \dfrac{3\pi }{4}=\cot x.....(i)$.

Now, we know, tangent function is negative in the second quadrant. So, the value of $\tan \dfrac{3\pi }{4}$ is equal to $-1$ . We will substitute the value of $\tan \dfrac{3\pi }{4}$ as $-1$ in equation $(i)$. On substituting the value of $\tan \dfrac{3\pi }{4}$ as $-1$ in equation $(i)$, we get:

$\cot x=-1$.

We know, the range for principal value is $\left( 0,\pi \right)$. So, we have to find a value of $x$ such that $x\in \left( 0,\pi \right)$ and $\cot x=-1$. The only possible value which satisfies both conditions is $x=\dfrac{3\pi }{4}$.

So , the value of principal value of the inverse trigonometric function ${{\cot }^{-1}}\left( \tan \dfrac{3\pi }{4} \right)$ is equal to $\dfrac{3\pi }{4}$.

Note: While solving the problem, make sure that the value of the inverse trigonometric function lies in the principal value range, i.e. $\left( 0,\pi \right)$for \[cot\] function. Students generally forget this condition and end up getting a wrong answer. So, this condition must be satisfied by the obtained principal value.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE