Alright. Thank you very much for your help in advance. I went thru all these problems myself and had questions on these particular ones. They shouldn't be too bad.

1) Convert X^2-y^2=1 to Polar...

.... so it will = (r^2)(cos^2theta) - (r^2)(sin^2theta) =1

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

where from here?

Mr F says: Use a double angle formula ....
2) Convert r=3sintheta to cartesian.

would x^2 + y^2 = 3y be correct?

Mr F says: Yes. Myself, I'd then write it in the form x^2 + (y - k)^2 = r^2 ....
3) What is the area of r^2=4cos2theta ??

Mr F says: See equations (13) - (15) of Lemniscate -- from Wolfram MathWorld (you should think about why integrate from -pi/4 to pi/4).
4) What is the area of the region that lies inside the first curve and outside the second curve

curve one = r=3costheta. curve 2 = r= 1+costheta

Mr F says: Done by Galactus.
5) What is the area of the region that lies inside both curves

1.st = r=sin2theta second = r=sintheta.

Mr F says: Draw a diagram. Think about how Galactus did the previous one.
6) Find the solutions of the equations (now into complex number stuff) ..

2x^2 -2x +1 =0

Mr F says: Use the quadratic formula.
7) Converting from complex numbers to polar form...

does -3+3i = (Sqrt(18))cis(tan(-1))

Mr F says: Draw an Argand diagram. The argument is clearly 3pi/4.
8) lastly, can someone provide a problem which wants me to convert from polar to complex? like... how rcostheta = a and risintheta = b

Mr F says: You mean convert from polar form to Cartesian form. Go to any textbook that covers this material. Or use google. [snip]