Polar coordinates, curves and areas

Learning objectives

Convert between Cartesian and polar coordinates using x = r cos θ, y = r sin θ, r² = x² + y² and tan θ = y/x, taking care to place θ in the correct quadrant, and convert the equations of curves between the two systems

Sketch a curve given as r = f(θ) over a stated interval of θ, marking where r is zero or greatest, noting any lines of symmetry and any values of θ for which r would be negative

Find the polar equations of the tangents to a curve that are parallel to the initial line by maximising r sin θ, and of those perpendicular to it by maximising r cos θ

Calculate the area of a sector bounded by a polar curve and two radii using ½∫r² dθ, choosing limits that sweep the region exactly once

Find the area between two polar curves, or between a curve and a line, by subtracting sectors or by splitting the region at the values of θ where the curves intersect