To convert the polar equation r = 8 sin(8θ) cos(θ) into a Cartesian equation, we can use the relationships between polar and Cartesian coordinates: x = r cos(θ) and y = r sin(θ).
First, let’s rewrite the polar equation:
r = 8 sin(8θ) cos(θ)
Now, we know that:
sin(8θ) = rac{y}{r},
cos(θ) = rac{x}{r}
Substituting these into our equation gives:
r = 8 imes rac{y}{r} imes rac{x}{r}
This simplifies to:
r² = 8xy
Since r² = x² + y² in Cartesian coordinates, we can substitute this in:
x² + y² = 8xy
This is the Cartesian equation for the given polar curve. Rearranging it gives a clearer form:
x² - 8xy + y² = 0
Therefore, the Cartesian equation for the curve r = 8 sin(8θ) cos(θ) is:
x² - 8xy + y² = 0