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8.2: The Tangent Function

( \newcommand{\kernel}{\mathrm{null}\,}\)

Let

A={π2+nπ:nZ}

and D=RA. Let

t:(π2,π2)R

be the inverse of the arctangent function. Note that t is increasing and differentiable on (π2,π2). We may extend t to a function on D as follows: For any xD, let

g(x)=sup{n:nZ,π2+nπ<x}

and define T(x)=t(xg(x)π).

Definition

With the notation of the above discussion, for any xD, we call the value T(x) the tangent of x, which we denote tan(x).

Proposition 8.2.1

The tangent function has domain D (as defined above), range R, and is differentiable at every point xD. Moreover, the tangent function is increasing on each interval of the form

(π2+nπ,π2+nπ),

nZ, with

tan((π2+nπ)+)=

and

tan((π2+nπ))=+.

Proof

These results follow immediately from our definitions. Q.E.D.

Definition

Let ER. We say a function f:ER is periodic if there exists a real number p>0 such that, for each xE,x+pE and f(x+p)=f(x). We say p is the period of a periodic function f if p is the smallest positive number for which f(x+p)=f(x) for all xE.

Proposition 8.2.2

The tangent function has period π.

Proof

The result follows immediately from our definitions. Q.E.D.

Proposition 8.2.3

(Addition formula for tangent)

For any x,yD with x+yD,

tan(x+y)=tan(x)+tan(y)1tan(x)tan(y).

Proof

Suppose y1,y2(π2,π2) with y1+y2(π2,π2). Let x1=tan(y1) and x2=tan(y2). Note that if x1>0, then x1x21 would imply that

x21x1,

which in turn implies that

y1+y2=arctan(x1)+arctan(x2)arctan(x1)+arctan(1x1)=π2,

contrary to our assumptions. Similarly, if x1<0, then x1x21 would imply that

x21x1,

which in turn implies that

y1+y2=arctan(x1)+arctan(x2)arctan(x1)+arctan(1x1)=π2,

contrary to our assumptions. Thus we must have x1x2<1. Moreover, suppose u is a number between x1 and x2. If x1>0, then

x2<1x1,

and so

u<1x1.

If x1<0, then

x2>1x1,

and so

u>1x1.

Now let

x=x1+x21x1x2.

We want to show that

arctan(x)=arctan(x1)+arctan(x2),

which will imply that

tan(y1)+tan(y2)1tan(y1)tan(y2)=tan(y1+y2).

We need to compute

arctan(x)=arctan(x1+x21x1x2)=x1+x21x1x2011+t2dt.

Let

t=φ(u)=x1+u1x1u,

where u varies between x1, where t=0, and x2, where t=x. Now

φ(u)=(1x1u)(x1+u)(x1)(1x1u)2=1+x21(1x1u)2,

which is always positive, thus showing that φ is an increasing function, and

11+t2=11+(x1+u1x1u)2=(1x1u)2(1x1u)2+(x1+u)2=(1x1u)2(1+x21)(1+u2).

Hence

arctan(x)=x2x111+u2du=0x111+u2du+x2011+u2du=x1011+u2du+arctan(x2)=arctan(x1)+arctan(x2)=arctan(x1)+arctan(x2).

Now suppose y1,y2(π2,π2) with y1+y2>π2. Then y1+y2(π2,π),x1>0,x2>0, and

x2>1x1.

With u and x as above, note then that as u increases from x1 to 1x1,t increases from 0 to +, and as u increases from 1x1 to x2,t increases from to x.

Hence we have

arctan(x)+π=x011+t2dt+011+t2dt++011+t2dt=x11+t2dt++011+t2dt=x21x111+u2du+1x1x111+u2du=x2x111+u2du=arctan(x2)arctan(x1)=arctan(x2)+arctan(x1).

Hence

tan(y1+y2)=tan(y1+y2π)=tan(arctan(x))=x1+x21x1x2=tan(y1)+tan(y2)1tan(y1)tan(y2).

The case when x1<0 may be handled similarly; it then follows that the addition formula holds for all y1,y2(π2,π2). The case for arbitrary y1,y2D with y1+y2D then follows from the periodicity of the tangent function. Q.E.D.


This page titled 8.2: The Tangent Function is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform.

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