Converting Between Systems

Calculus 2 · Axiom Academy

EXAMPLE Converting Between Systems Convert a point both ways between Cartesian and polar coordinates, checking the quadrant at every step. Part 1 — Cartesian → polar. Convert the point to polar coordinates . Part 2 — polar → Cartesian. Convert the point to Cartesian coordinates (x, y) . The conversion formulas: , , , and . Watch the quadrant. Both points plotted on a shared grid — each at the position fixed by its coordinates, with its polar ray and angle θ measured from the positive x-axis. Part 2 — Convert from polar to Cartesian Now run the conversion in the opposite direction. Nice work — you converted a point both directions and verified the quadrant each time. Here is what to carry forward: Cartesian → polar: and . Always check which quadrant the point is in. Polar → Cartesian: and — substitute and evaluate the trig directly. Quadrant checking is critical: a bare only returns angles in , so for x < 0 it lands a half-turn off; uses the signs of x and y to pick the right quadrant. Common values: knowing the key angles and their sines and cosines makes these conversions fast. These conversions are the foundation for calculus in polar coordinates — areas, arc lengths, and analyzing polar curves all build on them.

This is the written version of the interactive lesson above. See the full Calculus 2 course.