Precalculus

Trigonometry

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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:

tant=sintcost,cott=costsint,

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

cott=1tant.

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=π2+πn, nZ at which cost=0. The range of tant is all real numbers. Formally, we can write

dom(tant)={tR|tπ2+πn,nZ},codom(tant)=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 (π2,π2). Choose two arbitrary points t1,t2 from this interval such that t1>t2. To determine the sign of the difference tant1tant2, we represent it in the form

tant1tant2=sint1cost1sint2cost2=sint1cost2sint2cost1cost1cost2=sin(t1t2)cost1cost2.

The angles t1 and t2 lie in the 1st quadrant or in the 4th quadrant, where cosine is positive. Hence, cost1>0 and cost2>0. It's obvious that

0<t1t2<π,sin(t1t2)>0.

We see that if t1>t2, then

tant1tant2=sin(t1t2)cost1cost2>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:

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

where

Zeros of the Tangent Function

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

Graph of the Tangent Function

Since is undefined when the tangent function has vertical asymptotes at the points The graph of tangent consists of an infinite number of curves that can be obtained from each other by translation along the axis over where 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 is not defined at where The codomain (or range) of 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 and belong to the open interval Suppose that Determine the sign of the difference Using the sum-to-product identity, we can write

The angles and are in the quadrant or in the quadrant, where the sine is positive, so both and 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 when

Hence, the cotangent function is strictly decreasing in the interval

Parity of the Cotangent Function

The function 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

where

Zeros of the Cotangent Function

The cotangent will have zeros at the points where cosine is zero. Therefore, the solution of the equation is given by

Graph of the Cotangent Function

The cotangent graph has vertical asymptotes at the points It is always decreasing between the points of discontinuity. The cotangent function is periodic, so the separate curves can be obtained from each other using a translation by units, where

Graph of cotangent function
Figure 2.

See solved problems on Page 2.

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