Precalculus

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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.

Graph of tangent function
Figure 1.

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

Graph of cotangent function
Figure 2.

See solved problems on Page 2.

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