trig help....Polar to Rectangular?

Question

r= 4sin x

r^2= cos x

Answer 1

I assume you mean:

r = 4sinθ
and
r² = cosθ
_________

Remember the identities.

x = rcosθ
y = rsinθ
x² + y² = r²
_________

r = 4sinθ
r² = 4rsinθ
x² + y² = 4y
x² + y² - 4y = 0
x² + (y² - 4y + 4) = 4
x² + (y - 2)² = 4

This is a circle with center (0,2) and radius 2.
___________

r² = cosθ
r³ = rcosθ
(x² + y²)^(3/2) = x
____________

Answer 2

r^2 = 4rsin(theta)
y = rsin(theta)
r^2 = x^2 + y^2

x^2 + y^2 = 4y