Polar coordinates
Polar coordinates
- Polar coordinates give a point by distance $r$ from the origin and angle $\theta$.
- Links to Cartesian:
$$x = r\cos\theta, \qquad y = r\sin\theta, \qquad r^2 = x^2 + y^2$$
- Sector area: $\text{area} = \tfrac12\displaystyle\int r^2\,d\theta$.
Practice
In polar coordinates, the x-coordinate is given by:
x = r cos θ and y = r sin θ.
Practice
The area of a polar sector is ½∫r² dθ.
This is the standard polar area formula.
Converting curves
- Example: $r = 4\cos\theta$ → multiply by $r$: $r^2 = 4r\cos\theta$ → $x^2 + y^2 = 4x$ → $(x-2)^2 + y^2 = 4$, a circle radius $2$.
Practice
The polar curve r = 4cos θ is a circle. What is its radius?
r = 4cos θ gives x² + y² = 4x, i.e. (x−2)² + y² = 4 — a circle of radius 2.
You've got it
Key idea
- $x = r\cos\theta$, $y = r\sin\theta$, $r^2 = x^2 + y^2$
- polar sector area $= \tfrac12\int r^2\,d\theta$
- $r = 4\cos\theta$ is a circle of radius 2 centred at $(2, 0)$