Pre-Calculus: Complex Numbers - Trig or Polar Form

This lesson has not been reviewed.

Please purchase the lesson to review.

Applications of Trigonemtry
Complex Numbers in Trig Form
Expressing a Complex Number in Trigonometric or Polar Form Page [1 of 2]
Now suppose that we think about complex numbers or any complex number in a slightly different way. Well, how
should we write that? Let’s draw a picture. Suppose here’s a complex number. I’ll call it z and it’s x + yi . Now if I
connect that and make the vector sitting here, then what we can see is, if I formed this into a little right triangle, this
here is x and this here would be y . Let me call this angle ? . Now, the length of this, if I call that r , we know that’s
just in fact the modulus of this complex number. So r is just the modulus of z , which I remind you is just the square
root of the components squared. It’s the Pythagorean Theorem. It’s just the length of that vector.
But given all that, ? and r , how can I write r ? Well, I could use trigonometry to convert the x and the y to ? and
r . How would I do that specifically? Well, how could I get x ? Well, I could notice that in fact x is the adjacent side
to this angle, and so I see that cosine would probably be a prudent thing to consider. Let’s consider cos? . That
equals
adjacent
hypotenuse
, so
x
r
. And if I multiply through by r , I see that x=rcos? . Now if I look at the y , I see that’s
the opposite side, so let’s consider sine. So sin y
r
? = , which implies that y=rsin? . Okay, well that’s the value of
x and that’s the value of y , so if I insert that into this complex number z , I could replace x by just stuff in terms of
the r , the modulus, and this angle.
So if I do that, what do I see? Well, I’d see that z=rcos?+irsin?. And you’ll notice a common factor of r , so I
factor that out and what I’d see is z=r(cos?+isin?) . And this is known as the trig form, or sometimes it’s referred
to as the “polar” form, of a complex number. So if you take a complex number, x + iy , if that complex number makes
an angle of ? measure with the positive real axis, and the length of this vector is r , given by this, then it turns out you
can write z equivalently as r(cos?+isin?) , where we understand this is just the x part here and that’s the y , the
real part from the imaginary part right here.
Let’s do an example where we convert from this form to this form. Let’s try an example together. So let’s consider
(? 3,+i) and let’s write that in trigonometric form or in polar form. Let’s draw a picture of this and see what we’ve
got going on here. So ? 3 is over here, and then I go 1 unit up, so it looks like our point is somewhere over here.
This is z . Let me call this z . So this is z . And you can see that this point, if I connect it you can see the triangle.
And so this is length 1, this is ? 3 . We can use the Pythagorean Theorem to figure out the length of this by taking
this squared plus 1 squared. Well, (? 3)2 is just 3, plus 12 which is 1, so 3+1=4. But then I need to take the
square root in order to compute the modulus, and 4=2. So in fact this has length 2. So that’s the r part.
And you’ll notice in the formula, what do we need to find? We need to find r , and we just did that; that’s 2. Now we
need to find the angle we make with the positive real axis, and then we put it into this form and we get the polar form
or the trig form. So now I’ve got to figure out what this angle is right here. Well, if you look at this triangle and you see
that one leg has length squared of 3, one leg has length 1, one leg has length 2, then in fact this angle right in here
must be in fact, what? Well, this must be 30°. So I recognize this as a standard triangle that we should know. That’s
30° or another way of thinking of that is
6
?
. So if this is
6
?
, how can I find the whole thing? Well, the whole thing is
just ? radians and then I subtract the
6
?
. So this angle ? is the entire straight angle,
6
? ?? , which if you just
subtract is
5
6
?
, getting a common denominator over here of
6
6
.
Applications of Trigonemtry
Complex Numbers in Trig Form
Expressing a Complex Number in Trigonometric or Polar Form Page [2 of 2]
Well, that’s the angle. Well, if I know the angle and I know the modulus, then I can write z as this length here, 2,
times cosine of this angle,
5
6
?
, plus i times the sine of that angle
5
6
?
. So in fact this is the trigonometric or polar
form of this number.
Do you see how I found that? I found that by first graphing it. I first visually looked at it. And then I figured out the
length of this vector, and that gives me this r . And then I had to figure out that angle, that pitch, which I was able to
do using some basic trigonometry. And then I was able to write it in this form, which is identical—these two numbers
are the same but it’s just a different form. This way it tells me how far over to go in the real direction and how far to go
in the imaginary direction. This tells me what angle I should pitch at and then how far off—the 2 tells me how far out
to go. So it’s two different ways of representing the same point. This turns out to be really handy when you want to
start to perform some basic arithmetical functions with complex numbers. We’ll take a look at that coming up next. I’ll
see you there.