# Precalculus

## Trigonometry # 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:

$\tan t = \frac{{\sin t}}{{\cos t}},\;\;\cot t = \frac{{\cos t}}{{\sin t}},$

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

$\cot t = \frac{1}{{\tan t}}.$

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

$\text{dom}(\tan t) = \left\{ {t \in \mathbb{R} \left|\, t \ne \frac{\pi }{2} + \pi n \right.,\,n \in \mathbb{Z}} \right\},\;\text{codom}(\tan t) = \mathbb{R},$

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

$\tan {t_1} - \tan {t_2} = \frac{{\sin {t_1}}}{{\cos {t_1}}} - \frac{{\sin {t_2}}}{{\cos {t_2}}} = \frac{{\sin {t_1}\cos {t_2} - \sin {t_2}\cos {t_1}}}{{\cos {t_1}\cos {t_2}}} = \frac{{\sin \left( {{t_1} - {t_2}} \right)}}{{\cos {t_1}\cos {t_2}}}.$

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

$0 \lt {t_1} - {t_2} \lt \pi , \Rightarrow \sin \left( {{t_1} - {t_2}} \right) \gt 0.$

We see that if $$t_1 \gt t_2,$$ then

$\tan {t_1} - \tan {t_2} = \frac{{\sin \left( {{t_1} - {t_2}} \right)}}{{\cos {t_1}\cos {t_2}}} \gt 0,$

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:

$\tan \left( { - t} \right) = \frac{{\sin \left( { - t} \right)}}{{\cos \left( { - t} \right)}} = \frac{{ - \sin t}}{{\cos t}} = - \frac{{\sin t}}{{\cos t}} = - \tan t.$

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 $$\pi:$$

$\tan \left( {t + \pi n} \right) = \tan t,$

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

$t = \pi n,\;n \in \mathbb{Z}.$

### 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

$\text{dom}(\cot t) = \left\{ {t \in \mathbb{R} \left|\, t \ne \pi n \right.,\,n \in \mathbb{Z}} \right\},\;\text{codom}(\cot t) = \mathbb{R}.$

### 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

$\cot {t_1} - \cot {t_2} = - \frac{{\sin \left( {{t_1} - {t_2}} \right)}}{{\sin {t_1}\sin {t_2}}}.$

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,

$0 \lt {t_1} - {t_2} \lt \pi , \Rightarrow \sin \left( {{t_1} - {t_2}} \right) \gt 0.$

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:

$\cot \left( { - t} \right) = \frac{{\cos \left( { - t} \right)}}{{\sin \left( { - t} \right)}} = \frac{{\cos t}}{{ - \sin t}} = - \frac{{\cos t}}{{\sin t}} = - \cot t.$

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 $$\pi:$$

$\cot \left( {t + \pi n} \right) = \cot t,$

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

$t = \frac{\pi }{2} + \pi n,\;n \in \mathbb{Z}.$

### 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 $$n \in \mathbb{Z}.$$