5. Find the Cartesian form of the equation, r3 = 3r cosØ.
To find the Cartesian form of the equation r^3 = 3r cos(θ), we need to express it in terms of x and y coordinates.
First, recall the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ):
x = r * cos(θ)
y = r * sin(θ)
Now, let's express r^3 = 3r cos(θ) in terms of x and y:
r^3 = 3r cos(θ)
(r * r^2) = 3(r * cos(θ)) (since r^2 = r * r)
r^2 = 3cos(θ)
Since we know that r^2 = x^2 + y^2, we can replace r^2 in the equation:
x^2 + y^2 = 3cos(θ)
Now, using the relationship between x, y, and θ, we can replace cos(θ) with x/r:
x^2 + y^2 = 3(x/r)
By multiplying both sides of the equation by r^2:
r^2 * (x^2 + y^2) = 3xr^2
Finally, replacing r^2 with x^2 + y^2:
x^2 + y^2 = 3xr^2
This is the Cartesian form of the equation r^3 = 3r cos(θ) in terms of x and y.