Convert each of the following points into the specified coordinate system.
(a) (-4, 2 Π /3) into Cartesian coordinates.
(b) (-1,-1) into polar coordinates.
Solution
(a) transform (-4, 2 Π /3) into Cartesian coordinates.
This conversion is very easy. All we require to do is plug the points into the formulas.
So, in Cartesian coordinates this point is (2, -2 3).
(b) Convert or transform (-1,-1) into polar coordinates.
Let's first get r.
r = √(-1)2 + (-1)2
= √2
Now, let's get θ.
θ = tan-1 (-1/-1)
= tan-1 (1) = Π/4
Though this is not the correct angle, this value of θ is in the first quadrant and the point we have been specified is in the third quadrant. As noted above we can obtain the correct angle by adding π onto this. Hence, the actual angle is,
θ = Π/4 + Π = 5Π /4
So, in polar coordinates the point is (√2, 5π/4). Note also that we could have utilized the first θ that we got by using a negative r. In this type of case the point could as well be written in polar coordinates
as (-√2, π/4).