Phasor Mathematics

Compare these three mathematical series, one for e^x, one for cosx, and one for sinx:

$$e^x = 1+x+ \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + + \frac{x^6}{6!} + + \frac{x^7}{7!} ...$$

$$cos \ x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...$$

$$sin \ x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ...$$

Note similarities in the terms of these three series. Aside from signs, it would appear the cosine series contains all the even-powered terms of e^x and the sine series contains all the odd-powered terms of e^x. Leonhard Euler, the Swiss mathematician who lived from 1707 to 1783, found a connection between these three series which became known as Euler’s relation.

You too can find this same connection if you substitute jx for x in the first (e^x) series, and then multiply all terms in the sine series by j. Leave the cosine series unaltered. Remember that j = √{−1} and that j² = −1, j³ = −j, j⁴ = 1, j⁵ = j, etc.

A fascinating mathematical identity discovered by Leonhard Euler (1707-1783), regarded by some as the most beautiful equation in all of mathematics, relates five of mathematics’ fundamental constants together:

Use Euler’s relation to translate this identity into trigonometric terms, where the truth of this identity will become more evident.

If you have studied complex numbers, you know that the same complex quantity may be written in two different forms: rectangular and polar. Take for example the complex quantity [(√3)/2] j¹/₂. The following illustration shows this point located on the complex plane, along with its rectangular dimensions:

Next, we see the same point, on the same complex plane, along with its polar coordinates:

Written out, we might express the equivalence of these two notations as such:

$$\frac{\sqrt{3}}{2} + j \frac{1}{2} = 1 \angle \frac{\pi}{6}$$

Expressed in a more general form, the equivalence between rectangular and polar notations would look like this:

However, a problem with the “angle” symbol (∠) is that we have no standardized way to deal with it mathematically. We would have to invent special rules to describe how to add, subtract, multiply, divide, differentiate, integrate, or otherwise manipulate complex quantities expressed using this symbol. A more profitable alternative to using the “angle” symbol is shown here:

Explain why this equivalence is mathematically sound.