Question

How do you find the value of \[\cot {90^ \circ }\] ?

Answer 1

Hint: Given is a cotangent function. That is cot function is the reciprocal of tan function.
 \[\cot \theta = \dfrac{1}{{\tan \theta }}\]
Here since we know the direct value of this function but we will go from basics. We will use basic three trigonometric functions. First tan function and then rewriting it as the ratio of sin and cos function. This will simplify our answer. So let’s start!

Complete step-by-step answer:
Given to find the value of \[\cot {90^ \circ }\]
We know that \[\cot \theta = \dfrac{1}{{\tan \theta }}\]
But \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
So putting this is the above expression we get
 \[ \Rightarrow \cot \theta = \dfrac{1}{{\dfrac{{\sin \theta }}{{\cos \theta }}}}\]
On rearranging the terms
 \[ \Rightarrow \cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}\]
Now we are asked to find the value of cot for \[{90^ \circ }\] . So putting this angle we get
 \[ \Rightarrow \cot {90^ \circ } = \dfrac{{\cos {{90}^ \circ }}}{{\sin {{90}^ \circ }}}\]
Now we already know that, \[\cos {90^ \circ } = 0\] and \[\sin {90^ \circ } = 1\] . So substituting these values in above ratio we get,
 \[ \Rightarrow \cot {90^ \circ } = \dfrac{0}{1}\]
And 0 divided by anything is 0 only. So,
 \[ \Rightarrow \cot {90^ \circ } = 0\]
This is the final answer.
So, the correct answer is “0”.

Note: These are the trigonometric functions. They show the relation between three sides of a right angled triangle. There are basic three functions and other three are reciprocals of the basic. Here cot function is also having other identities in which there are angular changes and their respective values. Also cot function is having angular identities as for the sum or difference of two angles that is used to find the value of an angle that needs to be divided in two special angles.
 \[\cot \left( {A \pm B} \right) = \dfrac{{\cot A.\cot B \mp 1}}{{\cot B \pm \cot A}}\]