Question

Express r=2cos(theta)+2sin(theta) as a cartesian function.

Answer 1

The function is in polar coordinates.

When this is the case, to pass to rectagular (cartesian) coordinates you use:

x = r cos(theta)
y = r sin(theta)

Then,

x = [2cos(theta) + 2sin(theta)]cos(theta) =

= 2 [cos(theta)]^2 + 2sin(theta)cos(theta) = 2 [cos(theta)]^2 + sin(2theta)

y = [2cos(theta) + 2sin(theta)]sin(theta) =

= 2 cos(theta)sin(theta) + 2[sin(theta)]^2 = sin(2theta) + 2[sin(theta)]^2