To convert the polar equation r = 8 sin(θ) cos(θ) into Cartesian coordinates, we’ll use the relationships between polar and Cartesian coordinates. In polar coordinates, we have:
- x = r cos(θ)
- y = r sin(θ)
- r = √(x² + y²)
Substituting the definitions of sine and cosine, we can express the polar equation as:
r = 8 imes rac{y}{r} imes rac{x}{r}
This simplifies to:
r² = 8xy
Now, because r² = x² + y², we can replace r² in our equation:
x² + y² = 8xy
Next, we can rearrange this equation to express it in a more standardized form:
x² – 8xy + y² = 0
This equation represents the Cartesian form of the curve defined by the original polar equation. It is quadratic in both x and y, indicating that it describes a conic section.
To further analyze the properties of the curve, we can apply techniques to complete the square or find the discriminant if needed, but in essence, we have successfully translated the polar equation to a Cartesian one.