Question

Find 2 pairs of polar coordinates for each point with r > 0 and the other with r < 0. Express theta in radians.
(3, 4)

Answer 1

Well, the conversions we need to know are
\[r^{2} = x^{2} + y^{2}\]
This comes from pythagorean theorem, so it makes sense. The 2nd conversion we need is
\[\tan \theta = \frac{ y }{ x }\]This also makes sense because sin is usually used to refer to y coordinates and cosine to x coordinates. And of course tangent is sin divided by cos.

So a polar coordinate is in the form (r, theta). Using the two conversions above, you think you can come up with one of the points?

Answer 2

I've come up with r = 5. & 3/5 = cos which got me the polar coordinate of (5, 0.93). I used my calculator, but I'm not sure how to get the other coordinate where r< 0

Answer 3

So there are two ways to get a polar coordinate. But first we need to understand what the polar coordinate actually says. Basically, theta is the direction youre facing, and the radius is the distance you walk in that direction. So its like as if you were standign at the origin on a big graph and you faced right and walked 5 steps. Its kinda like that. So as I said, (5, 0.93) means facing the direction of 0.93, I walk 5 units. Now, the other way to get that exact same coordinate is to face in the complete opposite direction and walk backwards!



Now when we walk backwards, our r becomes negative. So we need to take 0.93, add pi radians (because thats how you face the opposite direction), and make the r negative. So one point is

(5, 0.93) the other is (-5, 0.93 + pi)