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### 19.2. Complex Numbers in Polar Form

In an earlier chapter we saw that a point could be located by polar coordinates, as well as by rectangular coordinates. Similarly, a complex number can be given in polar form as well as in rectangular form.

Why another form? We will see that while addition and subtraction of complex numbers are best done in rectangular form, multiplication and division are easier in polar form.

##### Figure 19.5. Polar form of a complex number shown on a complex plane.

Figure 19-5 shows how the rectangular and polar forms are related. There we have plotted the complex number a + bi. If we now connect that point to the origin by a line segment of length r, it makes an angle θ with the horizontal axis. We can express the same complex number in terms of r and θ, if we note that

So

a = r cos θ

Similarly,

b = r sin θ

Substituting gives

We can also write this expression using the simpler notation,

 Named for the Swiss mathematician Leonhard Euler (1707–1783).
 The derivation of Euler's formula can be found in Chapter 31 of the calculus version of this book.

A third polar ...

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