How do you find a polar equation for the curve represented by the Cartesian equation xy = 1?

To convert the Cartesian equation xy = 1 into a polar equation, we start by using the relationships between Cartesian and polar coordinates. In polar coordinates, we have:

• x = r cos(θ)
• y = r sin(θ)

Substituting these into the Cartesian equation gives:

(r cos(θ))(r sin(θ)) = 1

This simplifies to:

r² cos(θ) sin(θ) = 1

We can use the identity sin(2θ) = 2 sin(θ) cos(θ) to rewrite the equation:

r² (1/2) sin(2θ) = 1

Multiplying both sides by 2, we have:

r² sin(2θ) = 2

Finally, to express this in terms of r, we solve for r:

r² = rac{2}{sin(2θ)}

Or:

r = rac{ ext{sqrt}(2)}{ ext{sqrt}(sin(2θ))}

This gives us the polar equation corresponding to the Cartesian equation xy = 1.