Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

Verified step by step guidance

1. Recall the relationship between polar and Cartesian coordinates: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r\) is the radius and \(\theta\) is the angle parameter.
2. Given the polar equation \(r = f(\theta)\), substitute \(r\) into the Cartesian coordinate formulas to express \(x\) and \(y\) in terms of \(\theta\).
3. Write the parametric equations as \(x(\theta) = f(\theta) \cdot \cos(\theta)\) and \(y(\theta) = f(\theta) \cdot \sin(\theta)\).
4. Note that \(\theta\) serves as the parameter that varies, typically within an interval such as \([0, 2\pi]\), to trace the curve in the Cartesian plane.
5. Thus, the polar equation \(r = f(\theta)\) is expressed in parametric form as \(\left(x(\theta), y(\theta)\right) = \left(f(\theta) \cos(\theta), f(\theta) \sin(\theta)\right)\).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a radius and an angle (r, θ) from the origin. The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential to convert between polar and Cartesian forms.

Parametric Equations

Parametric equations express coordinates as functions of a parameter, often denoted t or θ. Instead of y as a function of x, both x and y are defined separately in terms of the parameter, allowing more flexible representations of curves, including those defined in polar form.

Conversion from Polar to Cartesian Coordinates

To convert polar equations to Cartesian form, use the relationships x = r cos(θ) and y = r sin(θ). When r is given as a function of θ, substituting r = f(θ) into these formulas yields parametric equations x(θ) and y(θ) in Cartesian coordinates.