Question

Find the exact value of \[{\tan ^{ - 1}}\left( { - \dfrac{1}{{\sqrt 3 }}} \right)\] .

Answer 1

Hint: Inverse tan is the inverse function of the trigonometric function ‘tangent’. It is used to calculate the angle by applying the tangent ratio of the angle which is the opposite side divided by the adjacent side of the right triangle. For obtaining an expression for the definite integral of the inverse tan function, the derivative is integrated and the value at one point is fixed.

Complete step-by-step answer:
The inverse of “tan” is restricted to quadrants 1 and 4. It is the inverse function of the trigonometric function tangent and “tan” is negative in quadrants 2 and 4.
An inverse function or anti function is a function that reverses another function. Inverse trigonometric functions are also called “arc functions”. For a given value of a trigonometric function they produce the length of arc needed to obtain that particular value.
We know that,
 \[{\tan ^{ - 1}}\left( { - x} \right) = - {\tan ^{ - 1}}\left( x \right)\]
Hence,
 \[{\tan ^{ - 1}}\left( { - \dfrac{1}{{\sqrt 3 }}} \right) = - {\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right)\]
Now we can find the value of \[{\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right)\] and add a negative sign to get the value of \[ - {\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right)\] .
Let us assume \[{\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right)\] as “x”. Therefore, we have
 \[
   \Rightarrow \dfrac{1}{{\sqrt 3 }} = \tan \left( x \right) \\
   \Rightarrow \tan \left( {\dfrac{\pi }{6}} \right) = \tan \left( x \right) \;
 \]
As we know that \[\tan \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{{\sqrt 3 }}\] .
Now, comparing both the equations we have,
 \[ \Rightarrow x = \dfrac{\pi }{6}\]
We need to find the value of \[{\tan ^{ - 1}}\left( { - \dfrac{1}{{\sqrt 3 }}} \right) = - {\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right)\] ,
Since, \[{\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right) = x\] so,
\[{\tan ^{ - 1}}\left( { - \dfrac{1}{{\sqrt 3 }}} \right) = - x\]
 \[ \Rightarrow - x = - \dfrac{\pi }{6}\]
Hence the value of \[{\tan ^{ - 1}}\left( { - \dfrac{1}{{\sqrt 3 }}} \right)\] is \[ - \dfrac{\pi }{6}\]
So, the correct answer is “ \[ - \dfrac{\pi }{6}\] ”.

Note: For a given value of trigonometric function they produce the length of arc needed to obtain that particular value. The range of an inverse function is defined as the range of values of the inverse function that can attain within the defined domain of the function. The domain of a function is defined as the set of every possible independent variable where the function exists.