**Euler's formula** is a famous formula discovered by Leonhard Euler. It provides a link between complex algebra and trigonometry. It states that, for any angle *θ*,

- $ e^{\theta i} = \cos{\theta} + i \sin{\theta} $

Famously, this gives rise to Euler's identity:

- $ e^{\pi i} + 1 = 0 \therefore e^{\pi i} = -1 $

Multiplying a point in the complex plane by Euler's formula results in rotation of that point by an angle *θ*. This is the rationale for the theta function.

## TheoremEdit

Euler's formula works because of its link to trigonometry and its application to the complex plane.

Consider a right triangle, as shown here:The trigonometric ratios tell us:

- $ \cos{\theta} = \frac{adjacent}{hypotenuse} = \frac{a}{c} $
- $ \sin{\theta} = \frac{opposite}{hypotenuse} = \frac{b}{c} $

If you allow point *A* to lie at the origin of the complex plane (0, 0), then point *B* lies on the real axis and point *C* lies on the imaginary axis. Thus, point *C* becomes the sum of the real part and the imaginary part; i.e. *C* = *a*+*bi*.

Since the trig ratios allow us to define the angle *θ*, we can express this as cos *θ* + *i* sin *θ*. This implies that side *a* = cos *θ* and side *b* = *i* sin *θ*, which fits nicely with the parameters we've described here.

In 1740, Euler collaborated with other mathematicians at the time to obtain his formula *e*^{θi} = cos *θ* + *i* sin *θ*, after John Bernoulli noted that

- $ \frac{1}{(1-x)^{2}} = \frac{1}{2} (\frac{1}{1-xi} + \frac{1}{1+xi} $and
- $ \int{\frac{dx}{1+ax}} = \frac{1}{a} \ln{(1+ax)} + C $

and Roger Cotes noted that

- $ xi = \ln{(\cos{x} + i \sin{x})} $

50 years later, Caspar Wessel described complex numbers as residing in the complex plane, allowing subsequent mathematicians to apply Euler's formula to the complex plane.

## Euler's Formula applied to *θ*Edit

Below is a list of formulas for *θ* radians.

- $ e^{0} = 1 $

- $ e^{\frac{\pi}{6}i} = (\frac{\sqrt{3}}{2} + \frac{1}{2}i) $

- $ e^{\frac{\pi}{4}i} = (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i) $

- $ e^{\frac{\pi}{3}i} = (\frac{1}{2} + \frac{\sqrt{3}}{2}i) $

- $ e^{\frac{\pi}{2}i} = i $

- $ e^{\frac{2 \pi}{3}i} = (-\frac{1}{2} + \frac{\sqrt{3}}{2}i) $

- $ e^{\frac{3 \pi}{4}i} = (-\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i) $

- $ e^{\frac{5 \pi}{6}i} = (-\frac{\sqrt{3}}{2} + \frac{1}{2}i) $

- $ e^{\pi i} = -1 $

- $ e^{\frac{7 \pi}{6}i} = (-\frac{\sqrt{3}}{2} - \frac{1}{2}i) $

- $ e^{\frac{5 \pi}{4}i} = (-\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i) $

- $ e^{\frac{4 \pi}{3}i} = (-\frac{1}{2} - \frac{\sqrt{3}}{2}i) $

- $ e^{\frac{3 \pi}{2}i} = -i $

- $ e^{\frac{5 \pi}{3}i} = (\frac{1}{2} - \frac{\sqrt{3}}{2}i) $

- $ e^{\frac{7 \pi}{4}i} = (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i) $

- $ e^{\frac{11 \pi}{6}i} = (\frac{\sqrt{3}}{2} - \frac{1}{2}i) $

- $ e^{2 \pi i} = e^{0} = 1 $