How do you convert rectangular coordinates to polar coordinates?

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 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)##