INSTRUCTOR: In this video, we're going to look at the Cartesian coordinates and change them into polar coordinates. So find the polar coordinates of the following points written as Cartesian coordinates. So we've got minus 2 root 3, 2, and 3 minus 1. OK. So this is type x, y. Very, very important to draw a diagram. 

OK. So we want minus 2 root 3, which is roughly around there. And we want two units here. So we need to draw a line from here to here. And what we need to do is find the length of the line and find out what this angle is here. That will give me the polar coordinates. 

So r, you can just get-- it's an application of Pythagoras theorem. So it'd be x squared plus y squared. So this will be the square root of minus 2 root 3 squared plus 2 squared. 2 squared is 4. Root 3 squared is 3. And 4 times 3 is 12, so square root of 12 plus 4. That's the square root of 16, which gives me 4. So the length of this line is 4. 

To find the angle-- might be good to find this angle here first and take it away from pi, because theta would be the same as doing pi minus alpha. So that'll be pi minus arctan of 2, the y value divided by the x value. Don't worry about the negative sign here. We can completely ignore the negative sign. This would take into account that. 

And that will give me-- so this will be arctan of 1 over root 3 once we've canceled it down, which gives me pi/6. So pi minus pi/6 gives me 5 pi pi by 6. And therefore, minus 2 root 3, 2 will have polar coordinates for 5 pi by 6. 

OK. Right. In the second case, we've got a 3 minus a 3, x, y. Again drawing a diagram is really, really important. OK. So we're going three along, minus 3 down. So we need to draw a line. We need to find r and this angle. So r will be x squared plus a y squared, which will be 3 squared plus minus 3 squared. 

So that will be 9 plus 9, which is root 18. 18 is 2 times 9. And the square root of 9 is 3. So this will be 3 root 2, simplifying using surds. Theta will be arctan of y/x. And this time, we can take into account the negative sign-- minus 3/3. And that will give me that theta is minus pi over 4, which is-- we expect the angle to be negative because it's this side. 

And therefore, 3 minus 3 has polar coordinates 3 root 2 comma minus pi/4. The expectation would be to give this in as an exact answer if possible. OK. So this has been a video to show you how to change Cartesian coordinates into polar coordinates. I hope you've understood, and I thank you very much for watching.