8.2: The Tangent Function
( \newcommand{\kernel}{\mathrm{null}\,}\)
Let
A={π2+nπ:n∈Z}
and D=R∖A. 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 x∈D, let
g(x)=sup{n:n∈Z,−π2+nπ<x}
and define T(x)=t(x−g(x)π).
With the notation of the above discussion, for any x∈D, we call the value T(x) the tangent of x, which we denote tan(x).
The tangent function has domain D (as defined above), range R, and is differentiable at every point x∈D. Moreover, the tangent function is increasing on each interval of the form
(−π2+nπ,π2+nπ),
n∈Z, with
tan((π2+nπ)+)=−∞
and
tan((π2+nπ)−)=+∞.
- Proof
-
These results follow immediately from our definitions. Q.E.D.
Let E⊂R. We say a function f:E→R is periodic if there exists a real number p>0 such that, for each x∈E,x+p∈E 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 x∈E.
The tangent function has period π.
- Proof
-
The result follows immediately from our definitions. Q.E.D.
(Addition formula for tangent)
For any x,y∈D with x+y∈D,
tan(x+y)=tan(x)+tan(y)1−tan(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 x1x2≥1 would imply that
x2≥1x1,
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 x1x2≥1 would imply that
x2≤1x1,
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+x21−x1x2.
We want to show that
arctan(x)=arctan(x1)+arctan(x2),
which will imply that
tan(y1)+tan(y2)1−tan(y1)tan(y2)=tan(y1+y2).
We need to compute
arctan(x)=arctan(x1+x21−x1x2)=∫x1+x21−x1x2011+t2dt.
Let
t=φ(u)=x1+u1−x1u,
where u varies between −x1, where t=0, and x2, where t=x. Now
φ′(u)=(1−x1u)−(x1+u)(−x1)(1−x1u)2=1+x21(1−x1u)2,
which is always positive, thus showing that φ is an increasing function, and
11+t2=11+(x1+u1−x1u)2=(1−x1u)2(1−x1u)2+(x1+u)2=(1−x1u)2(1+x21)(1+u2).
Hence
arctan(x)=∫x2−x111+u2du=∫0−x111+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+∫0−∞11+t2dt+∫+∞011+t2dt=∫x−∞11+t2dt+∫+∞011+t2dt=∫x21x111+u2du+∫1x1−x111+u2du=∫x2−x111+u2du=arctan(x2)−arctan(−x1)=arctan(x2)+arctan(x1).
Hence
tan(y1+y2)=tan(y1+y2−π)=tan(arctan(x))=x1+x21−x1x2=tan(y1)+tan(y2)1−tan(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,y2∈D with y1+y2∈D then follows from the periodicity of the tangent function. Q.E.D.