Editing Trigonometric functions (section)

=== 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>.