3 Convert the following Cartesian equations to polar equatio

3. Convert the following Cartesian equations to polar equations.

(a) x2 y2 = 1

(b) y = x2

4. Convert the following polar equations to Cartesian equations.

(a) r() = b sin()m cos()

(b) r() = a sin(2) 2(cos3 +sin3 )

Solution

3

a) x^2 - y^2 = 1

x = r cos theta , y = r sin theta

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

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

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

b) y = x^2

r sin theta = ( r cos theta )^2

r sin theta = r^2 cos^2 theta

r^2 cos^2 theta - r sin theta = 0