Question

What is the domain of cotangent?

Answer 1

Hint: First understand the meaning of cotangent and the function it represents. Consider the cotangent function as the ratio of cosine function to the sine function. Assume the argument as x radian. Now, to define the domain of the cotangent function, use the fact that the denominator of the function cannot be 0. So substitute the sine function equal to 0 and use the formula if $\sin x=0$ then $x=n\pi $ where $n\in $ integers. Consider the set of all the real numbers except these values of x to get the answer.

Complete step by step solution:
Here we have been asked to find the domain of cotangent. First we need to understand the term cotangent and domain.
Now in mathematics the term cotangent is a trigonometric function. There are six trigonometric functions namely: - sine, cosine, tangent, secant, cosecant and cotangent. Cotangent function is the ratio of cosine and the sine function also can be said as the inverse of the tangent function. Mathematically it is given as:
$\Rightarrow \cot x=\dfrac{\cos x}{\sin x}$
Now, the domain is the set of values of x for which the function is defined. As we can see that we have the sine function in the denominator of the cotangent function and we know that for a function to be defined its denominator must not be 0 for any value of x, so let us find for which values of x the sine function is 0.
$\Rightarrow \sin x=0$
We know that sine function is 0 for the integral multiples of $\pi $, so we get,
$\Rightarrow x=n\pi $, $n\in $ integers
Therefore, the domain of the cotangent function is $R-\left\{ n\pi ,n\in Z \right\}$ where R is the set of all real numbers and Z is the set of all integers.

Note: You must remember the domain values of all the six trigonometric functions which is further used in the chapter of inverse trigonometric function. The domain of cosecant function is the same as the domain of the cotangent function because cosecant function is the inverse of the sine function. Also note that the domain of tangent and secant function is \[R-\left\{ \left( 2n+1 \right)\dfrac{\pi }{2},n\in Z \right\}\] while the domain of sine and cosine function is the set of all real numbers (R).