The coordinates of P can be written in Cartesian form as but can be written as in polar form where is the distance from the origin and is the angle of rotation counter-clockwise from the positive -axis.
To convert from polar to Cartesian you can use the following:
Polar equations are usually given in the form so as varies from 0 to points are placed around the whole graph varying distance.
Sketching curves
Common curves are:
, a circle with centre origin radius , a half line making angle , a spiral starting at origin
, They are called Cardioids They are defined for all if so to make - this can be defined in some contexts but for A-level this is undefined)
Cardioids fall into 2 categories “egg” shaped and those with a “dimple”, egg shaped are convex curves, dimples are concave at . They are egg shaped when and dimple shaped if - This is proved by considering the number of tangents that are perpendicular to the initial line.
Area enclosed by a Polar Curve
The area enclosed by a polar curve and half lines where is measured in radians is given as:
Very useful trig identities for integration
Tangent to polar Curve
You can differentiate parametrically to get:
To find a tangent line to the parallel to the initial set:
To find a tangent perpendicular to the initial line set: