Section 1.3 - Polar Coordinates

Section 1.3 - Polar Coordinates

Introduction to Polar Coordinates

Polar coordinates provide an alternative way to represent points in the plane, using a distance from a fixed point (the pole) and an angle from a fixed direction (the polar axis). This system is especially useful for describing curves and regions that are circular or have radial symmetry.

• Pole: The fixed point (usually the origin) from which distances are measured.

• Polar Axis: The fixed direction (usually the positive x-axis) from which angles are measured.

• Polar Coordinates (r, θ): A point is represented by its distance r from the pole and angle θ from the polar axis.

Key Properties:

• If r > 0, the point (r, θ) lies in the same quadrant as θ.

• If r = 0, the point is at the pole for any value of θ.

• Angles are measured counterclockwise from the polar axis.

Multiple Representations in Polar Coordinates

Unlike Cartesian coordinates, a single point in polar coordinates can have many representations due to the periodic nature of angles and the sign of r.

• For example, the point $(1, \frac{\pi}{4})$ can also be written as $(1, \frac{9\pi}{4})$, $(-1, \frac{5\pi}{4})$, etc.

• General representations: $(r, \theta + 2\pi n)$ and $(-r, \theta + (2n+1)\pi)$, where n is any integer.

Connection Between Polar and Cartesian Coordinates

There are standard formulas to convert between polar and Cartesian coordinates:

• From Polar to Cartesian:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

• From Cartesian to Polar:

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Examples of Conversion

• Example 1: Convert $(2, \frac{\pi}{3})$ from polar to Cartesian coordinates.

  • $x = 2 \cos \frac{\pi}{3} = 2 \times \frac{1}{2} = 1$

  • $y = 2 \sin \frac{\pi}{3} = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}$

  • So, the Cartesian coordinates are $(1, \sqrt{3})$.

• Example 2: Represent the point $(1, -1)$ in polar coordinates.

  • $r = \sqrt{1^2 + (-1)^2} = \sqrt{2}$

  • $\theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}$

  • So, polar coordinates are $(\sqrt{2}, -\frac{\pi}{4})$.

Polar Curves

A polar curve is defined by an equation of the form $r = f(\theta)$. The graph consists of all points $(r, \theta)$ whose coordinates satisfy the equation.

• Example 3: The curve $r = 2$ represents a circle with center at the pole and radius 2.

• Example 4: The curve $\theta = 1$ represents a straight line at angle $\theta = 1$ radian from the polar axis.

• Example 5: The curve $r = 1 + \sin \theta$ is a cardioid. To sketch, plot $r$ for various values of $\theta$.

Symmetry in Polar Curves

Polar curves can exhibit symmetry about the polar axis, the pole, or the line $\theta = \frac{\pi}{2}$.

• If replacing $\theta$ with $-\theta$ leaves the equation unchanged, the curve is symmetric about the polar axis.

• If replacing $r$ with $-r$ or $\theta$ with $\theta + \pi$ leaves the equation unchanged, the curve is symmetric about the pole.

• If replacing $\theta$ with $\pi - \theta$ leaves the equation unchanged, the curve is symmetric about the line $\theta = \frac{\pi}{2}$.

Common Polar Curves

The following table summarizes some common polar curves, their equations, and examples:

Name	Equation	Example
Circle	$r = a$	Circle of radius $a$ centered at the pole
Line	$\theta = \alpha$	Line at angle $\alpha$ from the polar axis
Cardioid	$r = a(1 + \cos \theta)$	Heart-shaped curve
Lemniscate	$r^2 = a^2 \cos 2\theta$	Figure-eight shape
Rose	$r = a \cos n\theta$	Petal-shaped curve (number of petals depends on $n$)
Spiral	$r = a\theta$	Archimedean spiral

Calculus with Polar Coordinates

To find slopes and tangents to polar curves, we use derivatives with respect to $\theta$:

• Formulas:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

  • $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$

  • $\frac{dx}{d\theta} = \frac{dr}{d\theta} \cos \theta - r \sin \theta$

  • $\frac{dy}{d\theta} = \frac{dr}{d\theta} \sin \theta + r \cos \theta$

• Horizontal tangents: Occur where $\frac{dy}{d\theta} = 0$ and $\frac{dx}{d\theta} \neq 0$.

• Vertical tangents: Occur where $\frac{dx}{d\theta} = 0$ and $\frac{dy}{d\theta} \neq 0$.

Example: Tangents to the Cardioid

• For $r = 1 + \sin \theta$, find the slope of the tangent line at $\theta = \frac{\pi}{3}$.

• Find points where the tangent is horizontal or vertical by solving $\frac{dy}{d\theta} = 0$ or $\frac{dx}{d\theta} = 0$.

Additional info: The study notes include all major concepts from the provided section, with expanded academic context and examples for clarity.