Question

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

Answer 1

Hint: For finding the value of the trigonometric identity for an given value you need to be clear about the angle associated with it, for smaller values or say smaller angles like zero, thirty, forty five, sixty and ninety we know the values and these values are easy to learn, if any bigger values comes into play that means you need to solve it with the help of graph.

Formulae Used: \[\,\tan 30 = \dfrac{1}{{\sqrt 3 }}\]

Complete step by step answer:
The given question is \[{\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right)\]
Let us assume an angle \[\theta \] which satisfies the given value, and when we find the value of the assumed angle in comparison with the known trigonometric identity for a given angle, then simply by comparison we can obtain the assumed angle.
Equating the identity to the assumed angle we get;
\[
  {\tan ^{ - 1}}\left( {\dfrac{1}{{\sqrt 3 }}} \right) = \theta \\
  \tan \theta = \dfrac{1}{{\sqrt 3 }} \\
 \]
Now we know that,
\[\tan 30 = \dfrac{1}{{\sqrt 3 }}\]
So by comparing the both equations we can have the value of the assumed angle.
Here our required angle is \[{30^ \circ }\].

Note: In order to tackle such a question you have to remember the values of trigonometric identities at different values of \[\theta \]. If you don’t remember the values then it can be a tough task to go through the question. You can also make a triangle if the sides of the triangle are given.
Trigonometric values are very easy to learn and are inter-relatable, like you only have to learn for \[\sin \theta \] rest all values can be found easily by using\[\sin \theta \]. \[\cos \theta \] Values are opposite to \[\sin \theta \] values. \[\tan \theta \] Values are ratios of \[\sin \theta \] to\[\cos \theta \], and \[\cot \theta \] is reciprocal of\[\tan \theta \]. Similarly \[\sec \theta \] and \[\cos ec\theta \] are reciprocal of \[\cos \theta \] and\[\sin \theta \].