Question

How do you find the domain and range for $cot \theta$?

Answer 1

Hint: We can think of the cot function as the inverse tangent function or we can think of it as the fraction of sine and cosine. Keeping those in mind we can find the range and domain of the cot function. We need to be aware of its graph as well. By seeing the graph it can easily be figured out what the range and domain are for the function of cotangent.

Complete step-by-step solution:
We have $cot\theta$. We are required to find the domain and range of this. The domain of any function is the set of points on which the function is defined. The range of the function means all the values that the function takes when the values of the domain are plugged in. Now cotangent is the inverse of the tangent function, we can say that:
$\cot \theta =\dfrac{1}{\tan \theta }$
Now, whenever the tangent function will be 0, the function of cotangent will not be defined. And we know that at $n\pi$, the tangent function turns to 0. Hence, the domain of the cot function will exclude these points. So, all the real numbers except the integral multiples of $\pi$, will form the domain of the function. In mathematical terms, we can say that the domain of $cot\theta$ is:
$\mathbb{R}-n\pi$, where $n\in\mathbb{Z}$
Now, we find the range of the cot function. For this we say that tangent takes all values of the real line. So, the range of cot will also be all the real numbers or $\mathbb{R}$. It would be more clear from the graph below.

Hence, the domain and range has been found out.

Note: You can think of the domain as all the possible $x$ values and the range as the set of all the possible $y$ values. It is quite common to mix the two up, so be very careful about that. Also, the range and domain of easy trigonometric functions like this should be known beforehand.