INSTRUCTOR: Hi. In this video, we're going to convert polar coordinates into Cartesian coordinates. So find the Cartesian coordinates of the following points written as polar coordinates. This first one is 5, pi by 6, and the second one is 9, minus 2 pi by 3. 

Now, the secret for doing this is the first number here represents r, the length of the line. And the angle here represents the angle from the positive x-axis measured clockwise and anticlockwise from minus 180 to-- sorry, minus pi to pi in radians. 

Now, the secret for doing this type of question is to draw a diagram. The first one is straightforward. You might find this useful for writing in South Asian Tennis Club-- Sine positive, All positive, Tangent positive, and Cosine is positive. 

OK, so we need a angle here of pi by 6. So the angles always start from this axis here. They go around up to here to pi and then around here to minus pi. So pi by 6 is here. It's the same as 30 degrees. It doesn't have to be an accurate angle. Draw a straight line of length p. And the x-coordinate we get by doing r cos theta because that's what it would be. It would be r cos of this angle, so that's going to be 5. Cos pi by 6. 

Now, we're OK in the first quadrant because everything is positive. So 5 cos pi by 6. And cos of pi by 6 is root 3 over 2 expected to use the exact form. So it will be 5 root 3 over 2 will be the x-coordinate. And then to find the y-coordinate, r sine theta-- this one is sine theta. That will be 5 sine pi by 6. Sine of pi by 6 is 1/2. So it's 5 times 1/2, which gives me 5 over 2. 

Therefore, 5, pi over 6 has Cartesian coordinates 5 pi over 2, 5 over 2. Let's now do the second one, which is 9, minus 2 pi by 3. This is even more important that you do a diagram on this one. So it's r theta is the format. So draw in a diagram. 

So we're going to do minus pi by 3. So that'll be around from here. So knowing your angles in radians will be this and a length of line 9 units. OK, it's a good idea to work out what this angle would be here for when we work out the use of the angle. 

So x will be r cos theta. So that will be 9 cos of pi by 3. But because cosine is negative in that quadrant, we need to put a minus sign in front, which will be minus 9 1/2, which will give me minus 9 over 2. And here we have y is equal to r sine theta. Again, cosine is actually negative. So it will be minus 9 sine pi by 3, which is minus 9. Sine of pi by 3 is root 3 over 2. So that will be minus 9 root 3 over 2. 

Therefore, 9 minus 2 pi over 3 has Cartesian coordinates minus 9 over 2, minus 9 root 3 over 2. And don't forget, here, we would expect both the coordinates here to be negative when they come out as the final answer. OK, so this has been a video to show you how to change polar coordinates into Cartesian coordinates. In the next video, we'll look at doing it the other way around. I hope you've understood. And I thank you very much for watching.