# Graphs of Tangent and Cotangent Functions

## Definitions of Tangent and Cotangent

Recall that the tangent and cotangent functions are defined in terms of the sine and cosine:

where \(t\) is the angle between the radius-vector of the point on the unit circle and the positive *x*-axis (measured counterclockwise).

We see from the definitions that cotangent is the reciprocal of tangent, that is

Let us consider the properties of these two functions in more detail.

## The Graph and Properties of the Tangent Function

### Domain and Codomain of the Tangent Function

The tangent function is not defined at the points \(t = \frac{\pi }{2} + \pi n,\) \(n \in \mathbb{Z}\) at which \(\cos t = 0.\) The range of \(\tan t\) is all real numbers. Formally, we can write

where **dom** denotes the domain and **codom** denotes the codomain or range of the function.

### Monotonicity of the Tangent Function

Show that the tangent function is increasing in the open interval \(\left( { - \frac{\pi }{2},\frac{\pi }{2}} \right).\) Choose two arbitrary points \(t_1, t_2\) from this interval such that \(t_1 \gt t_2.\) To determine the sign of the difference \(\tan {t_1} - \tan {t_2},\) we represent it in the form

The angles \(t_1\) and \(t_2\) lie in the \(1\text{st}\) quadrant or in the \(4\text{th}\) quadrant, where cosine is positive. Hence, \(\cos {t_1} \gt 0\) and \(\cos {t_2} \gt 0.\) It's obvious that

We see that if \(t_1 \gt t_2,\) then

which means that the tangent function is strictly increasing in the given interval.

### Parity of the Tangent Function

It is easy to prove that the tangent function is odd:

As any other odd function, the graph of tangent function is symmetric about the origin.

### Periodicity of the Tangent Function

The tangent function is periodic with the least period \(\pi:\)

where \(n \in \mathbb{Z}.\)

### Zeros of the Tangent Function

The tangent will be zero wherever its numerator (the sine) is zero. Hence, the roots of the equation \(\tan t = 0\) are given by

### Graph of the Tangent Function \(y = \tan t\)

Since \(\tan t\) is undefined when \(\cos t = 0,\) the tangent function has vertical asymptotes at the points \(t = \frac{\pi }{2} + \pi n,\) \(n \in \mathbb{Z}.\) The graph of tangent consists of an infinite number of curves that can be obtained from each other by translation along the \(x-\)axis over \(n\pi\) where \(n\) is an integer.

## The Graph and Properties of the Cotangent Function

### Domain and Codomain of the Cotangent Function

The function \(\cot t = \frac{{\cos t}}{{\sin t}}\) is not defined at \(t = \pi n,\) \(n \in \mathbb{Z}\) where \(\sin t = 0.\) The codomain (or range) of \(\cot t\) is the set of all real numbers. Thus, we have

### Monotonicity of the Cotangent Function

The cotangent is a decreasing function between any two adjacent points of discontinuity. We prove this as follows. Let points \(t_1\) and \(t_2\) belong to the open interval \(\left( {0,\pi } \right).\) Suppose that \(t_1 \gt t_2.\) Determine the sign of the difference \(\cot {t_1} - \cot {t_2}.\) Using the sum-to-product identity, we can write

The angles \(t_1\) and \(t_2\) are in the \(1\text{st}\) quadrant or in the \(2\text{nd}\) quadrant, where the sine is positive, so both \(\sin {t_1} \gt 0\) and \(\sin {t_2} \gt 0.\) Besides that,

Thus, the fraction in the right-hand side of the above formula is positive. Given a minus sign before the fraction, we conclude that \(\cot {t_1} - \cot {t_2} \lt 0\) when \(t_1 - t_2 \gt 0.\)

Hence, the cotangent function is strictly decreasing in the interval \(\left( {0,\pi } \right).\)

### Parity of the Cotangent Function

The function \(\cot t\) is odd:

The graph of cotangent function is symmetric about the origin.

### Periodicity of the Cotangent Function

Similarly to tangent, the cotangent function is periodic with the least period \(\pi:\)

where \(n \in \mathbb{Z}.\)

### Zeros of the Cotangent Function

The cotangent will have zeros at the points where cosine is zero. Therefore, the solution of the equation \(\cot t = 0\) is given by

### Graph of the Cotangent Function \(y = \cot t\)

The cotangent graph has vertical asymptotes at the points \(t = \pi n,\) \(n \in \mathbb{Z}.\) It is always decreasing between the points of discontinuity. The cotangent function is \(\pi\) periodic, so the separate curves can be obtained from each other using a translation by \(n\pi\) units, where \(n \in \mathbb{Z}.\)