What is a Cartesian equation for r equals 4sinx 6cosx?

Answer 1

To convert the polar equation ( r = 4\sin(x) + 6\cos(x) ) to Cartesian coordinates, we use the relationships ( r = \sqrt{x^2 + y^2} ), ( x = r\cos(x) ), and ( y = r\sin(x) ). Substituting ( \sin(x) = \frac{y}{r} ) and ( \cos(x) = \frac{x}{r} ) into the equation, we get:

[ r = 4\left(\frac{y}{r}\right) + 6\left(\frac{x}{r}\right). ]

Multiplying through by ( r ) leads to ( r^2 = 4y + 6x ). Replacing ( r^2 ) with ( x^2 + y^2 ) gives the Cartesian equation:

[ x^2 + y^2 = 4y + 6x. ]