Editing Euler's formula (section)

==== Relationship to trigonometry ====
[[File:Sine Cosine Exponential qtl1.svg|thumb|Relationship between sine, cosine and exponential function]]
Euler's formula, the definitions of the trigonometric functions and the standard identities for exponentials are sufficient to easily derive most trigonometric identities. It provides a powerful connection between [[mathematical analysis|analysis]] and [[trigonometry]], and provides an interpretation of the sine and cosine functions as [[weighted sum]]s of the exponential function:
<math display="block">\begin{align}
\cos x &= \operatorname{Re} \left(e^{ix}\right) =\frac{e^{ix} + e^{-ix}}{2}, \\
\sin x &= \operatorname{Im} \left(e^{ix}\right) =\frac{e^{ix} - e^{-ix}}{2i}.
\end{align}</math>

The two equations above can be derived by adding or subtracting Euler's formulas:
<math display="block">\begin{align}
e^{ix}  &= \cos x + i \sin x, \\
e^{-ix} &= \cos(- x) + i \sin(- x) = \cos x - i \sin x
\end{align}</math>
and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments {{mvar|x}}. For example, letting {{math|1=''x'' = ''iy''}}, we have:
<math display="block">\begin{align}
\cos iy &= \frac{e^{-y} + e^y}{2} = \cosh y, \\
\sin iy &= \frac{e^{-y} - e^y}{2i} = \frac{e^y - e^{-y}}{2}i = i\sinh y.
\end{align}</math>

In addition
<math display="block">\begin{align}
\cosh ix &= \frac{e^{ix} + e^{-ix}}{2} = \cos x, \\
\sinh ix &= \frac{e^{ix} - e^{-ix}}{2} = i\sin x.
\end{align}</math>

Complex exponentials can simplify trigonometry, because they are mathematically easier to manipulate than their sine and cosine components. One technique is simply to convert sines and cosines into equivalent expressions in terms of exponentials sometimes called ''complex sinusoids''.<ref>{{Cite web |title=Complex Sinusoids |url=https://ccrma.stanford.edu/~jos/filters06/Complex_Sinusoids.html |access-date=2024-09-10 |website=ccrma.stanford.edu}}</ref> After the manipulations, the simplified result is still real-valued. For example:

<math display="block">\begin{align}
\cos x \cos y &= \frac{e^{ix}+e^{-ix}}{2} \cdot \frac{e^{iy}+e^{-iy}}{2} \\
 &= \frac 1 2 \cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\
 &= \frac 1 2 \bigg( \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} + \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} \bigg)\\
 &= \frac 1 2 \left( \cos(x+y) + \cos(x-y) \right).
\end{align}
</math>

Another technique is to represent sines and cosines in terms of the [[real part]] of a complex expression and perform the manipulations on the complex expression. For example:
<math display="block">\begin{align}
\cos nx &= \operatorname{Re} \left(e^{inx}\right) \\
 &= \operatorname{Re} \left( e^{i(n-1)x}\cdot e^{ix} \right) \\
 &= \operatorname{Re} \Big( e^{i(n-1)x}\cdot \big(\underbrace{e^{ix} + e^{-ix}}_{2\cos x } - e^{-ix}\big) \Big) \\
 &= \operatorname{Re} \left( e^{i(n-1)x}\cdot 2\cos x - e^{i(n-2)x} \right) \\
 &= \cos[(n-1)x] \cdot [2 \cos x] - \cos[(n-2)x].
\end{align}</math>

This formula is used for recursive generation of {{math|cos ''nx''}} for integer values of {{mvar|n}} and arbitrary {{mvar|x}} (in radians).

Considering {{math|cos ''x''}} a parameter in equation above yields recursive formula for [[Chebyshev polynomials]] of the first kind.

{{see also|Phasor#Arithmetic}}