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## Precalculus

### Course: Precalculus > Unit 3

Lesson 6: Polar form of complex numbers# Converting a complex number from polar to rectangular form

Given a complex number in polar form, we can convert that number to rectangular form and plot it on the complex plane. Created by Sal Khan.

## Want to join the conversation?

- What is the difference between a modulus and a magnitude?(3 votes)
- Nothing, they're synonyms.(5 votes)

- why does the magnitude (r) have to be the points in the X-axis? Is it because a positive 4 was only available in the x-axis in the 4th quadrant or is r always supposed be be the x-axis?(2 votes)
- We use both the real and imaginary components to calculate the magnitude. The formula is: sqrt(a^2+b^2) for a+bi. Notice that it looks like the Pythagorean theorem. That’s because it is the Pythagorean theorem being applied to find a distance.(1 vote)

- Hey, so in one of the exercises, part of the answer to a question, is pi/3, when asking to solve for
*italics_arctan(-5squareroot15/-5squareroot5)_italics*but when I put both numbers into my calculator, they are each different. My calculator was in radian mode. Does anyone know whats going on? Thanks(1 vote) - say you have the form 6(cos210degrees +i SIN210 degrees) what would that be in standard form(1 vote)

## Video transcript

- [Instructor] We are told,
consider the complex number Z, which is equal to the square root of 17 times cosine of 346 degrees
plus I sine of 346 degrees. And they ask us to plot Z
in the complex plane below. If necessary round the points coordinates to the nearest integer. So, I encourage you to pause this video and at least think about where we would likely
plot this complex number. All right. Now let's work through it together. So when you look at it like this, you can see that what's being attempted is a conversion from polar
form to rectangular form. And if we're thinking about polar form, we can think about the angle
of this complex number, which is clearly 346 degrees. And so, 346 degrees is about 14 degrees
short of a full circle. So, it would get us probably
something around there. And then we also see what the magnitude or the modulus of the complex
number is right over here. Square root of 17. Square root of 17 is a
little bit more than four 'cause four squared is 16. So if we go in this direction, let's see, that's gonna be
about one, two, three, four. We're gonna go right about there. So, if I were to just guess where this is going to put us, it's going to put us right around here, right around four minus one I. But let's actually (indistinct)
get a calculator out and see if this evaluates
to roughly four minus one I. So for the real part,
let's go 346 degrees. And we're gonna take the cosine of it. And then we're gonna multiply that times the square root of 17. So times 17 square root,
a little over four, which is equal to that. Actually, yes. The real part does look
almost exactly four. Especially, if we are rounding
to the nearest integer. It's a little bit more than four. And now let's do the imaginary part. So we have 346 degrees. And we're gonna take the sine of it. And we're going to multiply
that times the square root of 17 times 17 square root,
which is equal to, yep. If we were round to the nearest integer, it's about negative one. So, we get to this point right over here, which is approximately four minus I. And we are done.