Polar coordinates

What polar coordinates covers

Points located by a distance and a direction instead of across and up. The curves that result spiral and loop in ways cartesian equations describe awkwardly, and the area formula is built from thin sectors rather than thin strips. Core Pure content, examinable on both compulsory papers of 9FM0, and two lessons long.

The main ideas

• Points given as a distance and an angle, converted both ways by the standard relations.
• Polar equations turned into cartesian form by manufacturing r squared, r cos theta and r sin theta, usually by multiplying through by r.
• Sketching the standard families, circles, half-lines, cardioids and roses, from the values of r at the four compass angles.
• The range of the angle that traces a curve once, and the angles where the curve passes through the pole.
• Tangents parallel and perpendicular to the initial line, from the derivatives of r sin theta and r cos theta.
• Area as half the integral of r squared, with limits taken between consecutive zeros of r or at the intersections of two curves.

The results it turns on

x = r cos θ, y = r sin θ, and r² = x² + y²

converting between the two coordinate systems

A = ½∫r² dθ

the area swept out by a polar curve

cos²θ = (1 + cos 2θ)/2

the identity that makes the squared radius integrable

dy/dθ = 0 for a tangent parallel to the initial line

with dx/d theta = 0 giving the perpendicular ones

r = a(1 + cos θ)

the cardioid, the standard curve of the unit

Where it usually goes wrong

• The area formula squares the radius, so the double angle identities are needed immediately and the squaring has to happen before any integration.
• Tangents come from differentiating r sin theta or r cos theta, not r. Setting the derivative of r to zero answers a different question about the curve.
• An inverse tangent alone does not fix the quadrant, so plot the point before committing to an angle.

Where to start

Polar curves first, and spend the time on sketching, because the limits in the second lesson are chosen off the sketch. Areas second. Revise radians and the integration of squared trigonometric functions beforehand, since both are used from the first line of the area work.