Euler's formula

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.

Theorem
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 eθi = 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
Below is a list of formulas for θ radians.
 * $$e^{0} = 1$$


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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