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How can polar coordinates be converted into Cartesian coordinates?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

To convert polar coordinates $(r, θ)$ into Cartesian coordinates $(x, y)$ , you use the following trigonometric relationships:

$x = r \cos θ$
$y = r \sin θ$

In this system, $r$ represents the distance from the origin to a point $P$ , and $θ$ is the angle formed by the line segment $O P$ and the positive $x$ -axis. These formulas allow for the transformation of equations between the two systems, which is a fundamental technique in analytic geometry for simplifying the study of curves and geometric problems.1234