Final answer:
To convert the polar equation r=3+3 cosθ to rectangular coordinates, use the transformations x = r cosθ and y = r sinθ, and apply trigonometric identities to express x and y in terms of r and θ, then eliminate θ to get the equation purely in terms of x and y.
Step-by-step explanation:
To express the equation r=3+3 cosθ in rectangular coordinates, we make use of the polar to rectangular coordinate transformations. The relationships x = r cosθ and y = r sinθ allow us to convert polar coordinates to rectangular coordinates. By substituting these into our original equation, we get:
x = (3 + 3 cosθ) cosθ
y = (3 + 3 cosθ) sinθ
We can simplify these by applying trigonometric identities. Considering that cos2θ = (1 + cos(2θ)) / 2, and sinθ cosθ = (1 / 2)sin(2θ), the equations become:
x = 3 cosθ + 3 cos2θ = 3 cosθ + 3 * (1 + cos(2θ)) / 2
y = 3 sinθ cosθ = (3 / 2)sin(2θ)
Finally, isolate cosθ and sin(2θ) as:
cosθ = x / (3 + 3 cosθ)
sin(2θ) = 2y / 3
We can use these expressions to substitute back and eliminate θ from the equation, obtaining a relationship purely in terms of x and y.