Question

Find the value of \[\sqrt {{{\cot }^{ - 1}}( - \sqrt 3 )} \]?

Answer 1

Hint: Here you have to use the trigonometric value of\[\cot \theta \], at such \[\theta \] value which satisfies your question demand. Inverse property means when you equate two values opposite to equal sign then you can relate those values accordingly and meet your demand, this property will also use here,
For example: \[
  \cot \theta = 1 \\
  then, \\
  \theta = {\cot ^{ - 1}}(1) \\
 \]

Complete step by step answer:
 Given equation is \[\sqrt {{{\cot }^{ - 1}}( - \sqrt 3 )} \]
In this question we have to suppose an arbitrary value of \[\theta \]then we have to equate the equation except for root, then after some arrangement, we can finally get the value of the \[{\cot ^{ - 1}}( - \sqrt 3 )\] and then afterward we can apply root both side and can have our final answer.
Let, \[{\cot ^{ - 1}}( - \sqrt 3 ) = \theta \]
Solving further, find the value of \[\theta \]
\[
  then,\,\cot (\theta ) = - \sqrt 3 \\
  we\,know\,at\,\theta = \dfrac{\pi }{3} \\
  \cot (\dfrac{\pi }{3}) = \sqrt 3 \\
  so,\,\cot ( - \dfrac{\pi }{3}) = - \sqrt 3 \\
 \]
\[hence,\,{\cot ^{ - 1}}( - \sqrt 3 ) = \dfrac{\pi }{3}\]
Now, here we obtain the required value, and now simply we have to put this obtained value in our main equation and then apply root on both the side for the final answer,
Final solution is \[\sqrt {{{\cot }^{ - 1}}( - \sqrt 3 )} = \sqrt {\dfrac{\pi }{3}} \]
Formulae Used: Value of \[\cot (\dfrac{\pi }{3}) = \sqrt 3 \]
Additional Information: 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.

Note:
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\[\sin \theta \].