This is a follow-up post to yesterday's "What is CIS Notation?" post. In this post we explore why it is useful to write complex numbers in their trigonometric form. Here you will find reference to the trigonometric double-angle formulas, and sum-of-angle formulas. If you're not familiar with these, you'll just have to take our word for those steps!

Let's start with a complex number in trigonometric form: 4cis(30º). If we were to write this number out without using the "cis" shorthand, it would look like this:

4[cos(30º) + isin(30º)]

This evaluates to approximately 3.4641 + 2i

Now supposing we wanted to square this complex number. You know the process, right?

(3.4641 + 2i)² =

(3.4641 + 2i)(3.4641 + 2i) =

12 + 6.9282i + 6.9282i - 4 = 

8 + 13.8564i

Let's do the same thing, but using the trig form for the complex number.

{4[cos(30º) + isin(30º)]}² =

4[cos(30º) + isin(30º)] · 4[cos(30º) + isin(30º)] = 

16[cos²(30º) + isin(30º)cos(30º) + isin(30º)cos(30º) - sin²(30º)] = 

16[cos²(30º) - sin²(30º) +2isin(30º)cos(30º)] = 

Now here's where those double-angle formulas come in:

cos(2x) = cos²(x) - sin²(x)

sin(2x) = 2sin(x)cos(x)

Thus, our expression can be simplified to:

16[cos(60º) +isin(60º)] = 

16cis(60º)

Do you see what just happened there? When all is said and done, we ended up squaring the magnitude, and doubling the angle, and that was it!

As another example, let's take the complex number that we used in our previous blog post: 3 + 4i. If you wanted to square this, you would multiply it out as follows:

(3 + 4i)(3 + 4i) = 

9 + 12i + 12i - 16 = 

-7 + 24i

On the other hand, if we had the angle in cis notation: 5cis(53.16º), we can square the complex number by squaring the magnitude, and doubling the angle:

5cis(53.16º) = 25cis(106.32º), and we're done, just like that! That was a LOT quicker!

And, incidentally, this works for any cubing, raising to the fourth power, etc. To raise a complex number to the n^th power, raise its magnitude to the n^th, and multiply its angle by n. Now that's "power"ful! 

But suppose we aren't squaring a complex number - suppose we're multiplying two different complex numbers? What happens then?

Let's make this very general; we'll call the two complex numbers r₁cisA and r₂cisB. Their product is:

r₁(cosA + isinA) · r₂(cosB + isinB) = 

r₁r₂(cosAcosB + icosAsinB + isinAcosB - sinAsinB) = 

r₁r₂[(cosAcosB - sinAsinB) +  i(cosAsinB + sinAcosB)] =

r₁r₂[cos(A + B)* +  isin(A + B)*] =

r₁r₂cis(A + B)

* sum of angle formulas were used to obtain this result.

Wow! Isn't that a great result - to multiply two complex numbers, you multiply their magnitudes and add their angles! That's easy!

But the real power (that was a double pun, by the way) of this way of writing complex numbers is in taking roots of numbers.

And that, I think, will become the topic of one more follow-up blog post!