Editing Trigonometric functions (section)

==Definitions in analysis==
[[file:Trigonometrija-graf.svg|thumb|right|[[graph of a function|Graphs]] of sine, cosine and tangent]]
[[File:Taylorsine.svg|thumb|right|The sine function (blue) is closely approximated by its [[Taylor's theorem|Taylor polynomial]] of degree 7 (pink) for a full cycle centered on the origin.]]
[[File:Taylor cos.gif|thumb|Animation for the approximation of cosine via Taylor polynomials.]]
[[File:Taylorreihenentwicklung des Kosinus.svg|thumb|<math>\cos(x)</math> together with the first Taylor polynomials <math>p_n(x)=\sum_{k=0}^n (-1)^k \frac{x^{2k}}{(2k)!}</math>]]

[[G. H. Hardy]] noted in his 1908 work ''[[A Course of Pure Mathematics]]'' that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number.<ref name="Hardy">{{citation|first=G.H.|last=Hardy|title=A course of pure mathematics|year=1950|edition=8th|pages=432–438}}</ref> Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.

Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
* Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.<ref name="Hardy"/>
* By a power series, which is particularly well-suited to complex variables.<ref name="Hardy"/><ref name="WW">Whittaker, E. T., & Watson, G. N. (1920). A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. University press.</ref>
* By using an infinite product expansion.<ref name="Hardy"/>
* By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.<ref name="Hardy"/>
* As solutions of a differential equation.<ref name="BS">Bartle, R. G., & Sherbert, D. R. (2000). Introduction to real analysis (3rd ed). Wiley.</ref>

===Definition by differential equations===

Sine and cosine can be defined as the unique solution to the [[initial value problem]]:{{sfn|Bartle|Sherbert|1999|p=247}}
:<math>\frac{d}{dx}\sin x= \cos x,\ \frac{d}{dx}\cos x= -\sin x,\ \sin(0)=0,\ \cos(0)=1. </math>

Differentiating again, <math display="inline">\frac{d^2}{dx^2}\sin x = \frac{d}{dx}\cos x = -\sin x</math> and <math display="inline">\frac{d^2}{dx^2}\cos x = -\frac{d}{dx}\sin x = -\cos x</math>, so both sine and cosine are solutions of the same [[ordinary differential equation]]
:<math>y''+y=0\,.</math>
Sine is the unique solution with {{math|''y''(0) {{=}} 0}} and {{math|''y''′(0) {{=}} 1}}; cosine is the unique solution with {{math|''y''(0) {{=}} 1}} and {{math|''y''′(0) {{=}} 0}}.

One can then prove, as a theorem, that solutions <math>\cos,\sin</math> are periodic, having the same period. Writing this period as <math>2\pi</math> is then a definition of the real number <math>\pi</math> which is independent of geometry.

Applying the [[quotient rule]] to the tangent <math>\tan x = \sin x / \cos x</math>, 
:<math>\frac{d}{dx}\tan x = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = 1+\tan^2 x\,,</math>
so the tangent function satisfies the ordinary differential equation
:<math>y' = 1 + y^2\,.</math>
It is the unique solution with {{math|''y''(0) {{=}} 0}}.

===Power series expansion===
The basic trigonometric functions can be defined by the following power series expansions.<ref>Whitaker and Watson, p 584</ref> These series are also known as the [[Taylor series]] or [[Maclaurin series]] of these trigonometric functions:
:<math>
\begin{align}
\sin x & = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\[6mu]
& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]
\cos x & = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\[6mu]
& = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}.
\end{align}
</math>
The [[radius of convergence]] of these series is infinite. Therefore, the sine and the cosine can be extended to [[entire function]]s (also called "sine" and "cosine"), which are (by definition) [[complex-valued function]]s that are defined and [[holomorphic]] on the whole [[complex plane]].

Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to [[meromorphic function]]s, that is functions that are holomorphic in the whole complex plane, except some isolated points called [[zeros and poles|poles]]. Here, the poles are the numbers of the form <math display="inline">(2k+1)\frac \pi 2</math> for the tangent and the secant, or <math>k\pi</math> for the cotangent and the cosecant, where {{mvar|k}} is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the [[Taylor series]] of the other trigonometric functions. These series have a finite [[radius of convergence]]. Their coefficients have a [[combinatorics|combinatorial]] interpretation: they enumerate [[alternating permutation]]s of finite sets.<ref>Stanley, Enumerative Combinatorics, Vol I., p. 149</ref>

More precisely, defining
: {{mvar|U<sub>n</sub>}}, the {{mvar|n}}th [[up/down number]],
: {{mvar|B<sub>n</sub>}}, the {{mvar|n}}th [[Bernoulli number]], and
: {{mvar|E<sub>n</sub>}}, is the {{mvar|n}}th [[Euler number]],
one has the following series expansions:<ref>Abramowitz; Weisstein.</ref> 
: <math>
\begin{align}
\tan x & {} = \sum_{n=0}^\infty \frac{U_{2n+1}}{(2n+1)!}x^{2n+1} \\[8mu]
& {} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 2^{2n} \left(2^{2n}-1\right) B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
& {} = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + \frac{17}{315}x^7 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>

: <math>
\begin{align}
\csc x &= \sum_{n=0}^\infty \frac{(-1)^{n+1} 2 \left(2^{2n-1}-1\right) B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
&= x^{-1} + \frac{1}{6}x + \frac{7}{360}x^3 + \frac{31}{15120}x^5 + \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>

: <math>
\begin{align}
\sec x &= \sum_{n=0}^\infty \frac{U_{2n}}{(2n)!}x^{2n}
= \sum_{n=0}^\infty \frac{(-1)^n E_{2n}}{(2n)!}x^{2n} \\[5mu]
&= 1 + \frac{1}{2}x^2 + \frac{5}{24}x^4 + \frac{61}{720}x^6 + \cdots, \qquad \text{for } |x| < \frac{\pi}{2}.
\end{align}
</math>

: <math>
\begin{align}
\cot x &= \sum_{n=0}^\infty \frac{(-1)^n 2^{2n} B_{2n}}{(2n)!}x^{2n-1} \\[5mu]
&= x^{-1} - \frac{1}{3}x - \frac{1}{45}x^3 - \frac{2}{945}x^5 - \cdots, \qquad \text{for } 0 < |x| < \pi.
\end{align}
</math>

===Continued fraction expansion===

The following [[continued fraction]]s are valid in the whole complex plane:

:<math> \sin x =
\cfrac{x}{1 + \cfrac{x^2}{2\cdot3-x^2 +
\cfrac{2\cdot3 x^2}{4\cdot5-x^2 +
\cfrac{4\cdot5 x^2}{6\cdot7-x^2 + \ddots}}}}</math>

:<math> \cos x = \cfrac{1}{1 + \cfrac{x^2}{1 \cdot 2 - x^2 + \cfrac{1 \cdot 2x^2}{3 \cdot 4 - x^2 + \cfrac{3 \cdot 4x^2}{5 \cdot 6 - x^2 + \ddots}}}}</math>

:<math>\tan x = \cfrac{x}{1 - \cfrac{x^2}{3 - \cfrac{x^2}{5 - \cfrac{x^2}{7 - \ddots}}}}=\cfrac{1}{\cfrac{1}{x} - \cfrac{1}{\cfrac{3}{x} - \cfrac{1}{\cfrac{5}{x} - \cfrac{1}{\cfrac{7}{x} - \ddots}}}}</math>

The last one was used in the historically first [[proof that π is irrational]].<ref>{{citation|editor1-last = Berggren|editor1-first = Lennart|editor2-last = Borwein|editor2-first = Jonathan M.|editor2-link = Jonathan M. Borwein| editor3-last = Borwein|editor3-first = Peter B.|editor3-link = Peter B. Borwein|last = Lambert|first = Johann Heinrich|orig-year = 1768|chapter = Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques|title = Pi, a source book|place = New York|publisher = [[Springer Science+Business Media|Springer-Verlag]] |year = 2004|edition = 3rd|pages = 129&ndash;140|isbn = 0-387-20571-3}}</ref>

===Partial fraction expansion===

There is a series representation as [[partial fraction expansion]] where just translated [[Multiplicative inverse|reciprocal function]]s are summed up, such that the [[Pole (complex analysis)|pole]]s of the cotangent function and the reciprocal functions match:<ref name="Aigner_2000"/>
: <math>
\pi \cot \pi x = \lim_{N\to\infty}\sum_{n=-N}^N \frac{1}{x+n}.
</math>
This identity can be proved with the [[Gustav Herglotz|Herglotz]] trick.<ref name="Remmert_1991"/>
Combining the {{math|(–''n'')}}th with the {{math|''n''}}th term lead to [[absolute convergence|absolutely convergent]] series:
:<math>
\pi \cot \pi x = \frac{1}{x} + 2x\sum_{n=1}^\infty \frac{1}{x^2-n^2}.
</math>
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
:<math>
\pi\csc\pi x = \sum_{n=-\infty}^\infty \frac{(-1)^n}{x+n}=\frac{1}{x} + 2x\sum_{n=1}^\infty \frac{(-1)^n}{x^2-n^2},
</math>
:<math>\pi^2\csc^2\pi x=\sum_{n=-\infty}^\infty \frac{1}{(x+n)^2},</math>
:<math>
\pi\sec\pi x = \sum_{n=0}^\infty (-1)^n \frac{(2n+1)}{(n+\tfrac12)^2 - x^2},
</math>
:<math>
\pi \tan \pi x = 2x\sum_{n=0}^\infty \frac{1}{(n+\tfrac12)^2 - x^2}.
</math>

===Infinite product expansion===
The following infinite product for the sine is due to [[Leonhard Euler]], and is of great importance in complex analysis:<ref>Whittaker and Watson, p 137</ref>
:<math>\sin z = z \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2 \pi^2}\right), \quad z\in\mathbb C.</math>
This may be obtained from the partial fraction decomposition of <math>\cot z</math> given above, which is the logarithmic derivative of <math>\sin z</math>.<ref>Ahlfors, p 197</ref> From this, it can be deduced also that
:<math>\cos z = \prod_{n=1}^\infty \left(1-\frac{z^2}{(n-1/2)^2 \pi^2}\right), \quad z\in\mathbb C.</math>

=== Euler's formula and the exponential function ===
[[File:Sinus und Kosinus am Einheitskreis 3.svg|thumb|<math>\cos(\theta)</math> and <math>\sin(\theta)</math> are the real and imaginary part of <math>e^{i\theta}</math> respectively.]]

[[Euler's formula]] relates sine and cosine to the [[exponential function]]:
:<math> e^{ix}  = \cos x + i\sin x.</math> 
This formula is commonly considered for real values of {{mvar|x}}, but it remains true for all complex values.

''Proof'': Let <math>f_1(x)=\cos x + i\sin x,</math> and <math>f_2(x)=e^{ix}.</math> One has <math>df_j(x)/dx= if_j(x)</math> for {{math|1=''j'' = 1, 2}}. The [[quotient rule]] implies thus that <math>d/dx\, (f_1(x)/f_2(x))=0</math>. Therefore, <math>f_1(x)/f_2(x)</math> is a constant function, which equals {{val|1}}, as <math>f_1(0)=f_2(0)=1.</math> This proves the formula.

One has
:<math>\begin{align}
e^{ix}  &= \cos x + i\sin x\\[5pt]
e^{-ix}  &= \cos x - i\sin x.
\end{align}</math>

Solving this [[linear system]] in sine and cosine, one can express them in terms of the exponential function:
: <math>\begin{align}\sin x &= \frac{e^{i x} - e^{-i x}}{2i}\\[5pt]
\cos x &= \frac{e^{i x} + e^{-i x}}{2}.
\end{align}</math>

When {{mvar|x}} is real, this may be rewritten as 
: <math>\cos x = \operatorname{Re}\left(e^{i x}\right), \qquad \sin x = \operatorname{Im}\left(e^{i x}\right).</math>

Most [[List of trigonometric identities|trigonometric identities]] can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity <math>e^{a+b}=e^ae^b</math> for simplifying the result.

Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of [[topological group]]s.<ref>{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Topologie generale |publisher=Springer |year=1981|at=§VIII.2}}</ref> The set <math>U</math> of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group <math>\mathbb R/\mathbb Z</math>, via an isomorphism
<math display="block">e:\mathbb R/\mathbb Z\to U.</math>
In pedestrian terms <math>e(t) = \exp(2\pi i t)</math>, and this isomorphism is unique up to taking complex conjugates.

For a nonzero real number <math>a</math> (the ''base''), the function <math>t\mapsto e(t/a)</math> defines an isomorphism of the group <math>\mathbb R/a\mathbb Z\to U</math>. The real and imaginary parts of <math>e(t/a)</math> are the cosine and sine, where <math>a</math> is used as the base for measuring angles. For example, when <math>a=2\pi</math>, we get the measure in radians, and the usual trigonometric functions. When <math>a=360</math>, we get the sine and cosine of angles measured in degrees.

Note that <math>a=2\pi</math> is the unique value at which the derivative 
<math display="block">\frac{d}{dt} e(t/a)</math> 
becomes a [[unit vector]] with positive imaginary part at <math>t=0</math>. This fact can, in turn, be used to define the constant <math>2\pi</math>.

=== Definition via integration ===
Another way to define the trigonometric functions in analysis is using integration.<ref name="Hardy"/><ref>{{citation|last=Bartle|year=1964|title=Elements of real analysis|publisher=|pages=315–316}}</ref> For a real number <math>t</math>, put
<math display="block">\theta(t) = \int_0^t \frac{d\tau}{1+\tau^2}=\arctan t</math>
where this defines this inverse tangent function. Also, <math>\pi</math> is defined by
<math display="block">\frac12\pi = \int_0^\infty \frac{d\tau}{1+\tau^2}</math>
a definition that goes back to [[Karl Weierstrass]].<ref>{{cite book |last=Weierstrass |first=Karl |author-link=Karl Weierstrass |chapter=Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter Betrag zwischen zwei gegebenen Grenzen liegt |trans-chapter=Representation of an analytical function of a complex variable, whose absolute value lies between two given limits |language=de |title=Mathematische Werke |volume=1 |publication-place=Berlin |publisher=Mayer & Müller |year=1841 |publication-date=1894 |pages=51–66 |chapter-url=https://archive.org/details/mathematischewer01weieuoft/page/51/ }}</ref>

On the interval <math>-\pi/2<\theta<\pi/2</math>, the trigonometric functions are defined by inverting the relation <math>\theta = \arctan t</math>. Thus we define the trigonometric functions by
<math display="block">\tan\theta = t,\quad \cos\theta = (1+t^2)^{-1/2},\quad \sin\theta = t(1+t^2)^{-1/2}</math>
where the point <math>(t,\theta)</math> is on the graph of <math>\theta=\arctan t</math> and the positive square root is taken.

This defines the trigonometric functions on <math>(-\pi/2,\pi/2)</math>. The definition can be extended to all real numbers by first observing that, as <math>\theta\to\pi/2</math>, <math>t\to\infty</math>, and so <math>\cos\theta = (1+t^2)^{-1/2}\to 0</math> and <math>\sin\theta = t(1+t^2)^{-1/2}\to 1</math>. Thus <math>\cos\theta</math> and <math>\sin\theta</math> are extended continuously so that <math>\cos(\pi/2)=0,\sin(\pi/2)=1</math>. Now the conditions <math>\cos(\theta+\pi)=-\cos(\theta)</math> and <math>\sin(\theta+\pi)=-\sin(\theta)</math> define the sine and cosine as periodic functions with period <math>2\pi</math>, for all real numbers.

Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First,
<math display="block">\arctan s + \arctan t = \arctan \frac{s+t}{1-st}</math>
holds, provided <math>\arctan s+\arctan t\in(-\pi/2,\pi/2)</math>, since
<math display="block">\arctan s + \arctan t= \int_{-s}^t\frac{d\tau}{1+\tau^2}=\int_0^{\frac{s+t}{1-st}}\frac{d\tau}{1+\tau^2}</math>
after the substitution <math>\tau \to \frac{s+\tau}{1-s\tau}</math>. In particular, the limiting case as <math>s\to\infty</math> gives
<math display="block">\arctan t + \frac{\pi}{2} = \arctan(-1/t),\quad t\in (-\infty,0).</math>
Thus we have
<math display="block">\sin\left(\theta + \frac{\pi}{2}\right) = \frac{-1}{t\sqrt{1+(-1/t)^2}} = \frac{-1}{\sqrt{1+t^2}} = -\cos(\theta)</math>
and
<math display="block">\cos\left(\theta + \frac{\pi}{2}\right) = \frac{1}{\sqrt{1+(-1/t)^2}} = \frac{t}{\sqrt{1+t^2}} = \sin(\theta).</math>
So the sine and cosine functions are related by translation over a quarter period <math>\pi/2</math>.

===Definitions using functional equations===
One can also define the trigonometric functions using various [[functional equation]]s.

For example,<ref name="Kannappan_2009"/> the sine and the cosine form the unique pair of [[continuous function]]s that satisfy the difference formula
: <math>\cos(x- y) = \cos x\cos y + \sin x\sin y\,</math>
and the added condition
: <math>0 < x\cos x < \sin x < x\quad\text{ for }\quad 0 < x < 1.</math>

===In the complex plane===
The sine and cosine of a [[complex number]] <math>z=x+iy</math> can be expressed in terms of real sines, cosines, and [[hyperbolic function]]s as follows:
: <math>\begin{align}\sin z &= \sin x \cosh y + i \cos x \sinh y\\[5pt]
\cos z &= \cos x \cosh y - i \sin x \sinh y\end{align}</math>

By taking advantage of [[domain coloring]], it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of <math>z</math> becomes larger (since the color white represents infinity), and the fact that the functions contain simple [[Zeros and poles|zeros or poles]] is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

{| style="text-align:center"
|+ '''Trigonometric functions in the complex plane'''
|[[File:Trig-sin.png|thumb]]
<math>
\sin z\,
</math>
[[File:Trig-cos.png|thumb]]
<math>
\cos z\,
</math>
|[[File:Trig-tan.png|thumb]]
<math>
\tan z\,
</math>
[[File:Trig-cot.png|thumb]]
<math>
\cot z\,
</math>
|[[File:Trig-sec.png|thumb]]
<math>
\sec z\,
</math>
[[File:Trig-csc.png|thumb]]
<math>
\csc z\,
</math>
|}