How do you evaluate \[\cot (\dfrac{{3\pi }}{2})\].

Hint: As we all are very well aware of trigonometric functions that are sin, cos, tan, cot, etc. So, in this question, we have to deal with one of the trigonometric functions that are the cot function. We know that tan and cot function gives positive values for any angle which lies in the 1st and 3rd quadrant. Firstly we will try to convert our radian angle into degrees and then we will try to evaluate it. So to convert radians into degrees we use one simple formula that is. By using this \[\theta \, = \dfrac{{{{180}^ \circ }}}{\pi } \times radian\,angle\] formula we will try to evaluate it.

Complete step by step solution:
 To evaluate the value of \[\cot (\dfrac{{3\pi }}{2})\]
Firstly we will convert the angle from radian unit to degree units by using one formula that is
\[ \Rightarrow \] \[\theta \, = \dfrac{{{{180}^ \circ }}}{\pi } \times radian\,angle\]
Now putting the value of radian angle in the above formula
\[ \Rightarrow \theta \, = \dfrac{{{{180}^ \circ }}}{\pi } \times \dfrac{{3\pi }}{2}\]
By solving the above equation we get
\[ \Rightarrow \theta \, = 27{0^ \circ }\]
Now we have to find the value of \[\cot {270^ \circ }\]
And we know that \[\cot {270^ \circ } = 0\] because it is a standard value (\[\cot {270^ \circ }\], \[\cot {90^ \circ } = 0\])

So our final answer is 0.

Note:
By using the above formula we can easily convert radians into degrees for any angle. We can also generalise the value of the cot function that \[\cot {\left( {90 \times \left( {2n - 1} \right)} \right)^ \circ } = 0\] where value of n starts from 1. By using the above expression we can evaluate angle in terms of \[{90^ \circ }\] very easily.