Welcome back to Professor Baker’s Math Class! In this session, we are shifting our perspective from the traditional rectangular grid to the circular world of Polar Functions (Chapter 10, Sections 3 and 4). While Cartesian coordinates $(x, y)$ describe a location based on horizontal and vertical distances, polar coordinates $(r, \theta)$ describe a location based on distance from the origin and direction.

1. Understanding Polar Coordinates

In the polar system, every point is defined by:

  • $r$ (radius): The distance from the pole (origin).
  • $\theta$ (theta): The directed angle from the polar axis (positive x-axis).

As seen in the class notes, we can visualize these points by rotating an angle $\theta$ and marching out a distance $r$.

2. Coordinate Conversion

Being able to switch between systems is crucial. We use trigonometry to derive the connection between polar and Cartesian forms:

From Polar to Cartesian:
To find $x$ and $y$, we look at the components of the triangle formed by $r$ and $\theta$:
$$x = r \cos \theta$$
$$y = r \sin \theta$$

From Cartesian to Polar:
To find $r$ and $\theta$, we use the Pythagorean theorem and the tangent function:
$$r^2 = x^2 + y^2 \implies r = \sqrt{x^2+y^2}$$
$$\tan \theta = \frac{y}{x}$$

Example from notes: Converting the point $(2, \pi/3)$ results in $(1, \sqrt{3})$.

3. Graphing Polar Curves

Polar functions create beautiful, symmetrical shapes that would be very difficult to write as functions of $y$ and $x$. In our notes, we explored several specific curves:

  • Circles: Simple equations like $r = 2$.
  • Cardioids: Heart-shaped curves, such as $r = 1 + \sin \theta$.
  • Roses: Flower-like shapes, such as the four-leaved rose $r = \cos 2\theta$.

4. Calculus with Polar Curves

Just as we did with rectangular coordinates, we can apply calculus to find properties of these curves.

Area Under a Curve

Instead of summing rectangles, we sum infinite triangular sectors swept out by the radius. The formula for the area of a polar region is:

$$A = \int_{a}^{b} \frac{1}{2} r^2 \, d\theta$$

In one of our complex examples, we calculated the area inside the circle $r = 3 \sin \theta$ but outside the cardioid $r = 1 + \sin \theta$. This required finding the points of intersection (where $\sin \theta = 1/2$) and setting up the integral for the difference between the two curves.

Arc Length

To find the length of a curve in polar coordinates, we use a modified distance formula that accounts for the changing radius:

$$L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$$

For the cardioid $r = 1 + \sin \theta$, we found that the total length around the curve simplifies to exactly 8.

Tangents and Slopes

Finding the slope of the tangent line $\frac{dy}{dx}$ is a bit more involved because $y$ and $x$ are both functions of $\theta$. We use the product rule:

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

Keep practicing these conversions and integrals! Polar functions are a powerful tool for modeling circular and periodic motion. Check the attached PDF for the full step-by-step derivations.