Question

How do you find the value of \[\cot \left( {\dfrac{{5\pi }}{3}} \right)\] using the double angle or half angle identity?

Answer 1

Hint: We are to find the value of the cotangent function given to us in the question. So, to find this, we will use the property that cotangent function is the reciprocal of tangent function. So, we will replace the cotangent function and transform the given function completely into tangent function. Then we will find the value of tangent function for the given angle. Later, we will substitute the value in the trigonometric function and hence obtain our required value.

Complete step by step answer:
Given, The cotangent function is, \[\cot \left( {\dfrac{{5\pi }}{3}} \right)\]. Now, we know, cotangent function is the reciprocal of tangent function, that is,
$\tan x = \dfrac{1}{{\cot x}}$
$ \Rightarrow \cot x = \dfrac{1}{{\tan x}}$
Now, using this property in the given trigonometric function, we get,
\[\cot \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{\tan \left( {\dfrac{{5\pi }}{3}} \right)}}\]
Now, we have to find the value of \[\tan \left( {\dfrac{{5\pi }}{3}} \right)\].
Therefore, \[\tan \left( {\dfrac{{5\pi }}{3}} \right) = \tan \left( {\dfrac{{6\pi - \pi }}{3}} \right)\]
\[ \Rightarrow \tan \left( {\dfrac{{5\pi }}{3}} \right) = \tan \left( {2\pi - \dfrac{\pi }{3}} \right)\]
Therefore, the angle lies in the fourth quadrant, where the value of tangent function is negative.
Therefore, we can further simplify it as,
\[ \Rightarrow \tan \left( {\dfrac{{5\pi }}{3}} \right) = \tan \left( { - \dfrac{\pi }{3}} \right)\]
\[ \Rightarrow \tan \left( {\dfrac{{5\pi }}{3}} \right) = - \tan \left( {\dfrac{\pi }{3}} \right)\]
Now, we know the value of $\tan \left( {\dfrac{\pi }{3}} \right) = \sqrt 3 $.
Substituting this value, we get,
\[ \Rightarrow \tan \left( {\dfrac{{5\pi }}{3}} \right) = - \sqrt 3 \]
Now, substituting this value in the original equation, we get,
\[\cot \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{{\tan \left( {\dfrac{{5\pi }}{3}} \right)}} = \dfrac{1}{{ - \sqrt 3 }}\]
\[ \therefore \cot \left( {\dfrac{{5\pi }}{3}} \right) = - \dfrac{1}{{\sqrt 3 }}\]
Therefore, the value of \[\cot \left( {\dfrac{{5\pi }}{3}} \right)\] is $ - \dfrac{1}{{\sqrt 3 }}$.

Note: This problem can also be solved by directly substituting the value of cotangent function from the trigonometric table. First we have to find the function lying in which quadrant and then find the value that is repeated in that quadrant of the cotangent function. Then substitute the value for that quadrant with the appropriate sign for that quadrant.