Grades Pre-Calculus & Calculus Pre-CalculusPolar Coordinates & Parametric Equations

Polar Coordinates & Parametric Equations

🔭 Pre-Calculus

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Polar Coordinates & Parametric Equations

Represent curves using polar and parametric forms and convert between systems.

The polar coordinate system represents points by their distance r from the origin and angle θ from the positive x-axis. Parametric equations express x and y independently as functions of a third parameter t, allowing representation of curves that are not functions.

x = r\cos\theta \qquad y = r\sin\theta \qquad r = \sqrt{x^2+y^2} \qquad \theta = \arctan\left(\frac{y}{x}\right)

\text{Polar circle: } r = 2a\cos\theta \text{ (center } (a,0)\text{, radius } |a|\text{)}

\text{Cardioid: } r = a(1 + \cos\theta)

Parametric equations x = f(t), y = g(t) describe a curve traced as t varies. The slope of the curve at parameter t is dy/dx = (dy/dt)/(dx/dt), found by the chain rule.

\text{Parametric slope: } \frac{dy}{dx} = \frac{dy/dt}{dx/dt}

🌀Parametric Circle

x = 3cos(t), y = 3sin(t) for t ∈ [0, 2π] traces a circle of radius 3. Verify: x² + y² = 9cos²(t) + 9sin²(t) = 9.

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Remember This!

To eliminate the parameter, solve one equation for t and substitute into the other. For trig parametric equations, use the Pythagorean identity sin²t + cos²t = 1.

✏️ Try It!

Convert the polar point (4, π/3) to rectangular coordinates.

Take Quiz 📝 — 25 Questions