Converting Between Polar and Rectangular

Pre-Calculus · Axiom Academy

LESSON Converting Between Polar and Rectangular One point, two languages — and the two formulas that translate between rectangular (x, y) and polar . Drop a single point in the plane. Rectangular coordinates locate it by its horizontal and vertical offsets, (x, y) . Polar coordinates locate the same point by how far it is from the origin, r , and the angle its radius makes with the positive x -axis. Rectangular — horizontal and vertical offsets Polar — distance and direction 2. Polar → Rectangular: Project the Radius If you know , drop the point onto the axes. The radius is the hypotenuse of a right triangle; its horizontal leg is and its vertical leg is . That is all the conversion is — the two legs of the triangle. — how far right (or left) the radius reaches. — how far up (or down) the radius reaches. Valid for any angle and any r , including negative values — the signs of and handle every quadrant. Each gives exactly one (x, y) — this direction is the easy one. 3. Rectangular → Polar: Distance and Angle Going the other way you need two pieces. The distance r is the hypotenuse, straight from the Pythagorean theorem. The angle comes from the tangent ratio — but the calculator's only knows the ratio y/x , not which quadrant you're in. Worked example: (x, y) = (3, 4) , and since the point is in Quadrant I, . So the polar form is . One Pythagorean step — no quadrant worry. Always . The careful part — alone lands in Quadrants I or IV, so check the signs of x and y .

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