Question

How you evaluate $\cot \left( { - \dfrac{\pi }{6}} \right)?$

Answer 1

Hint:In this question, to find the cotangent of negative argument or angle, i.e. $\cot \left( { - \dfrac{\pi }{6}} \right)$, we have to use the formula for negative angles for tangent and cotangent, that is given as

$
\tan ( - x) = - \tan x \\
\cot ( - x) = - \cot x \\
$
Use this formula to convert the negative argument or angle into positive one for cotangent.

Complete step by step solution:
 In this question, we have to find the value of $\cot \left( { - \dfrac{\pi
}{6}} \right)$

As we know that cotangent and tangent of an argument becomes negative with equal magnitude when same argument also becomes negative, this can be understood as follows

Let us take value of $\tan x = y$, then the value of tangent when its argument that is $x$ becomes negative, can be given as
$\tan ( - x) = - \tan x$

And we know that $\tan x = y$,
$\therefore \tan ( - x) = - y$

Using this method in order to find the value of $\cot \left( { - \dfrac{\pi }{6}} \right)$
$ \Rightarrow \cot \left( { - \dfrac{\pi }{6}} \right) = - \cot \dfrac{\pi }{6}$

Now we have to put the value of $\cot \dfrac{\pi }{6}$ above in order to get the desired answer,

As we know that $\cot \dfrac{\pi }{6} = \sqrt 3 $

Putting this value above, we will get
$ \Rightarrow \cot \left( { - \dfrac{\pi }{6}} \right) = - \cot \dfrac{\pi }{6} = - \sqrt 3 $

Therefore we got the value of $\cot \left( { - \dfrac{\pi }{6}} \right) = - \sqrt 3 $ by evaluating it.

Note: Generally students do not remember the general values of secant, cosecant, and cotangent, but if you have remembered the general values of sine, cosine and tangent then you don’t have to worry about it, since sine and cosine are multiplicative inverse of each other similar case with cosine and secant, and tangent and cotangent.

 So you can write their values as follows $\csc x = \dfrac{1}{{\sin x}},\;\sec x = \dfrac{1}{{\cos x}}\;{\text{and}}\;\cot x = \dfrac{1}{{\tan x}}$

You can also solve this question by adding $2\pi $ to the argument and then finding the
content of the argument that comes after addition. Try this by yourself and check the answer.