Revision Points - Converting the Equation of a Polar Curve to Cartesian Form

Foundations of Polar and Cartesian Coordinates

Polar coordinates and Cartesian coordinates are two distinct systems used in geometry to specify points in a plane.
A point’s Cartesian coordinates (x, y) describe its location in relation to two perpendicular horizontal and vertical axes, while its polar coordinates (r, θ) describe its location in terms of distance from a fixed point (the pole) and an angle measured counter-clockwise from a fixed direction.
Understand that the relationship between both these coordinates is established by the equations x = rcos(θ) and y = rsin(θ).

Conversion Process

The conversion from polar coordinates to Cartesian form usually centres around replacing r, θ in the polar equation with corresponding expressions in x, y.
You need to remember the basic formulae: x = rcos(θ), y = rsin(θ), r² = x² + y², and tan(θ) = y/x.
Recognise that sometimes it will be necessary to use trigonometric identities to simplify the equation. For example, when the polar equation contains a cos(2θ) or sin(2θ), you may need to use double-angle formulas.
Be cautious when you are dealing with equations where r<0. Remember that negative r changes the direction of the line.

Examining the Cartesian Form

Be sure to simplify your equation after substitution to make it in the conventional Cartesian form.
The Cartesian equation may represent a variety of geometric shapes. These could be lines, circles, ellipses, parabolas, or hyperbolas.
Understand how to identify what shape a given equation represents. For instance, if both x and y are squared and the coefficients are equal, the Cartesian equation likely represents a circle.

Remember: The key to successful conversion is a fundamental understanding of both polar and Cartesian coordinates and the relationship between them. Practice converting between these forms and recognising the shapes represented will greatly strengthen your mathematical capabilities. It is also crucial to understand every concept thoroughly, as many concepts in Core Pure Math build upon each other.