Chapter 8: Polar Coordinates and Parametric Equations
Polar coordinates describe a point by its distance from the origin and angle from the positive x-axis. Parametric equations describe curves by expressing x and y separately in terms of a third variable t, enabling curves that can't be written as y = f(x) — circles, spirals, projectile paths.
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Sections
8.1Polar Coordinates
In polar coordinates, every point is (r, θ): r is the distance from the pole and θ is the angle from the polar axis. Unlike Cartesian coordinates, any point has infinitely many polar representations.
8.2Graphs of Polar Equations
Polar graphs are best understood by testing for symmetry, finding key points at θ = 0, π/2, π, 3π/2, and tracing how r changes. Rose curves, limaçons, and lemniscates each have distinctive shapes.
8.3Polar Form of Complex Numbers; De Moivre's Theorem
Complex numbers can be written as z = r(cosθ + i sinθ). Multiplication and division become geometric — multiply magnitudes and add angles. De Moivre's Theorem gives powers and roots of complex numbers directly.
8.4Plane Curves and Parametric Equations
Parametric equations express x and y separately as functions of a parameter t. This allows us to describe curves like circles, spirals, and projectile paths — curves that aren't functions of x alone — and adds the notion of direction and speed.
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