## Euler's Identity

Since is the algebraic expression of in terms of its rectangular coordinates, the corresponding expression in terms of its polar coordinates is

There is another, more powerful representation of in terms of its
polar coordinates. In order to define it, we must introduce *Euler's
identity*:

A proof of Euler's identity is given in the next chapter. Before, the only algebraic representation of a complex number we had was , which fundamentally uses Cartesian (rectilinear) coordinates in the complex plane. Euler's identity gives us an alternative representation in terms of polar coordinates in the complex plane:

*polar form*of the complex number , in contrast with the

*rectangular form*. Polar form often simplifies algebraic manipulations of complex numbers, especially when they are multiplied together. Simple rules of exponents can often be used in place of messier trigonometric identities. In the case of two complex numbers being multiplied, we have

A corollary of Euler's identity is obtained by setting to get

For another example of manipulating the polar form of a complex number, let's again verify , as we did above in Eq.(2.4), but this time using polar form:

*even*function ( ) while sine is

*odd*( ).

We can now easily add a fourth line to that set of examples:

Euler's identity can be used to derive formulas for sine and cosine in terms of :

Similarly, , and we obtain the following classic identities:

**Next Section:**

De Moivre's Theorem

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Elementary Relationships