Alternative Coordinates Summary

Trigonometry · Axiom Academy

SUMMARY Alternative Coordinates Let's review how polar and parametric representations provide powerful alternatives to Cartesian coordinates for circular motion and time-dependent paths. Definition: Points are represented as (r, θ) where r is the distance from the origin and θ is the angle from the positive x-axis Multiple Representations: Same point can be written in infinitely many ways: (r, θ) = (r, θ + 2πn) or (-r, θ + π) Natural Applications: Perfect for circular motion, radar systems, and any situation involving radial symmetry Angle Convention: Positive angles measured counterclockwise, negative angles clockwise from positive x-axis Definition: Express both x and y as functions of a third variable (parameter), typically t for time: x = f(t), y = g(t) Motion Description: Traces out a path as the parameter changes, naturally describing position over time Flexibility: Can represent curves that aren't functions in Cartesian form, including paths with loops or multiple y-values for one x Elimination: Convert to Cartesian by eliminating the parameter, though parametric form often gives more insight Converting Between Coordinate Systems Polar to Rectangular (3, π/6): Apply x = r cos θ and y = r sin θ to get x = 3 cos(π/6) = 3(√3/2) = 3√3/2 and y = 3 sin(π/6) = 3(1/2) = 3/2. Result: (3√3/2, 3/2) Rectangular to Polar (-2, 2√3): Calculate r = √((-2)² + (2√3)²) = √(4 + 12) = √16 = 4 Find the angle: tan θ = (2√3)/(-2) = -√3. Since point is in Quadrant II, θ = 2π/3

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