Expert: Steve Holleran - 5/9/2007

Question
change these rectanglular equations into polar equations
y=x^2

x^2-Y^2=1


change these polar equations into rectuangular
r^2 = sin 2theta
r=1+cos theta


thanks

Answer
Hi Patrick,

The conversion equations you need are :

x = r cos theta
y = r sin theta


x^2 + y^2 = r^2

theta = arctan(y/x)

Rectangular to polar:

y = x^2

r sin theta = (r cos theta)^2  = r^2 cos^2 theta

 sin theta = r cos^2 theta

 sin theta / cos theta = r

or, r = sec theta tan theta


x^2 - y^2 = 1

r^2 cos^2 theta - r^2 sin^2 theta = 1

r^2 (cos^2 theta - sin^2 theta) = 1

r^2 cos 2 theta = 1

r^2 = 1 / cos 2 theta = sec 2 theta

Polar to rectangular:

r^2 = sin 2 theta = 2 sin theta cos theta

x^2 + y^2 = 2 (y/r) (x/r)

x^2 + y^2 = 2xy / r^2 = 2xy / x^2 + y^2

or, (x^2 + y^2)^2 - 2xy = 0


r = 1 + cos theta

sqrt(x^2 + y^2) = 1 + (x/r)

r * sqrt(x^2 + y^2) = r + x

sqrt(x^2 + y^2) * sqrt(x^2 + y^2) = sqrt(x^2 + y^2) + x

(x^2 + y^2) - sqrt(x^2 + y^2) = x.

The polar to rectangular ones are pretty messy sometimes.

Hope this helps

Steve