Question

Convert r=1+3sin2x from a polar equation to a rectangular equation.

Answer 1

I assume you mean

r = 1 + 3 sin 2t
where t = theta

x = r cos t
y = r sin t

sin 2 t = 2 sin t cos t
so
r = 1 + 6 sin t cos t
r = 1 + 6 (y/r)(x/r)
r^3 = 1 + 6 x y
but r^2 = x^2+y^2
r= (x^2+y^2)^.5
so
(x^2+y^2)^(3/2) = 1 + 6 x y
(x^2+y^2)^3 = 1 + 12 x y + 36 x^2 y^2
you can multiply the left out

Answer 2

wouldnt the "1"in the 1+.... in the 4th step become r^2 when you multiply both sides by r^2.

Answer 3

You are right
r^3 = r^2 + 6 x y
(x^2+y^2)^3 = (x^2+y^2)^2 + 6 x y

Answer 4

x^6 + 3 x^4y^2 + 3 x^2 y^4 + y^6
= x^4 + 2 x^2y^2 + y^4 + 6 x y

Answer 5

Thanks a lot for your help