91Ó°ÊÓ

To convert the polar equation \(r\cos \theta = \sin 2\theta\) into its Cartesian form, we follow these steps: 1. Replace \(r\cos\theta\) with \(x\): \(x = \sin 2\theta\) 2. Replace \(\sin 2\theta\) with \(2\sin\theta\cos\theta\): \(x = 2\sin\theta\cos\theta\) 3. Replace \(\sin\theta\) in terms of \(y\) and \(r\): \(x = 2\left(\frac{y}{r}\right)\cos\theta\) 4. Replace \(r\) in terms of \(x\) and \(\cos\theta\): \(x = 2\left(\frac{y\cos\theta}{x}\right)\cos\theta\) 5. Simplify the expression: \(x = 2\left(\frac{y}{x}\right)(\cos\theta)^2\) 6. Replace \((\cos\theta)^2\) with \(1 - (\sin\theta)^2\): \(x = 2\left(\frac{y}{x}\right)(1 - (\sin\theta)^2)\) 7. Replace \((\sin\theta)^2\) with \(1 - (\cos\theta)^2\): \(x = 2\left(\frac{y}{x}\right)\left(1 - \frac{y^2}{x^2 + y^2}\right)\) 8. Simplify the expression: \(x(x^2 + y^2) = 2yx^2\) The resulting Cartesian equation is: \(x(x^2 + y^2) = 2yx^2\) This curve is a Cardioid with symmetry with respect to the x-axis.