Converting Between Systems
Key Questions
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It is usually suitable to use polar coordinates when you deal with round objects like circles, and to use rectangular coordinates when you deal with more straight edges like rectangles.
I hope that this was helpful.
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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 yourr andtheta into your equation forx=rcos theta andy=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 forr inr^2=x^2+y^2 to getr=sqrt(x^2+y^2) and solve fortheta intan theta= y/x to gettheta=arctan (y/x) (arctan is just tan inverse, ortan^-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 positivetheta COUNTERCLOCKWISE from the x-axis, even if ourr 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 andtheta= 2pi/3 intox=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 thatr must be positive andtheta must be between 0 andpi )So,
x=1 andy=1 . We can findr andtheta from:
r=sqrt(1^2+1^2)=sqrt2
theta=arctan (y/x)=arctan(1)=pi/4 The polar coordinate is
(sqrt2,pi/4)