To convert from polar to rectangular:
x=rcos theta
y=rsin theta
To convert from rectangular to polar:
r^2=x^2+y^2
tan theta= y/x
This is where these equations come from:
Basically, if you are given an (r,theta) -a polar coordinate- , you can plug your r and theta into your equation for x=rcos theta
and y=rsin theta to get your (x,y).
The same holds true for if you are given an (x,y)-a rectangular coordinate- instead. You can solve for r in r^2=x^2+y^2 to get r=sqrt(x^2+y^2) and solve for theta in tan theta= y/x to get theta=arctan (y/x) (arctan is just tan inverse, or tan^-1). Note that there can be infinitely many polar coordinates that mean the same thing. For example, (5, pi/3)=(5,-5pi/3)=(-5,4pi/3)=(-5,-2pi/3)...However, by convention, we are always measuring positive theta COUNTERCLOCKWISE from the x-axis, even if our r is negative.
Let's look at a couple examples.
( 1)Convert (4,2pi/3) into Cartesian coordinates.
So we just plug in our r=4 and theta= 2pi/3 into
x=4cos 2pi/3=-2
y=4sin 2pi/3=2sqrt3
The cartersian coordinate is (-2,2sqrt3)
(2) Convert (1,1) into polar coordinates. ( since there are many posibilites of this, the restriction here is that r must be positive and theta must be between 0 and pi)
So, x=1 and y=1. We can find r and theta from:
r=sqrt(1^2+1^2)=sqrt2
theta=arctan (y/x)=arctan(1)=pi/4
The polar coordinate is (sqrt2,pi/4)
