Converting between polar and cartesian form: rectangular form of a circle, polar form of a parabola

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Converting between polar and cartesian form: rectangular form of a circle, polar form of a parabola.

In the first example we convert x^2+y^2=9 to polar coordinates. We recognize this as a circle of radius 3 centered at the origin, but we proceed with the formal conversion. We use the conversion x=rcos(theta) and the conversion y=rsin(theta) to convert to polar coordinates. We apply the identity (cos(x))^2+(sin(x))^2=1 and simplify the polar equation to r^2=9. The interesting thing about this example is that there are two valid solutions to the polar equation: r=+3 and r=-3. Both of these polar equations trace out a circle of radius 3 centered at the origin -- they just start at different point.

In the second example, we convert the polar equation r=sec(theta)*tan(theta) to rectangular form. We use the identity r^2=x^2+y^2 together with the conversions x=rcos(theta) and y=rsin(theta) to manipulate the equation into the form y=x^2. In other words, this is the polar equation for a parabola passing through the origin!