Defining Polar Coordinates and Differentiating in Polar Form

#9.7 Defining Polar Coordinates and Differentiating in Polar Form

First of all, give yourself a pat on the back! You’ve made it through parametric equations (9.1, 9.2, and 9.3) and vector-valued functions (9.4 and 9.5) and tied them together in solving motion problems in 9.6. Where do we go from there?

The last section of this unit deals with polar coordinates. In our introduction, they are briefly defined as part of a two-dimensional coordinate system dealing with a line’s distance from the origin (r) and the angle said line makes with the positive x-axis (θ)… but what does that really mean?

To understand polar coordinates, we need to understand the functions that utilize them the most: polar functions!

#🐻‍❄️ What are Polar Functions?

Polar functions, also known as circular functions, are functions commonly graphed in a polar coordinate system, which uses a distance (r) from a fixed point, known as the pole, and an angle (θ)) measured counter-clockwise from the positive x-axis, to determine the coordinates of a point. These functions are often used in physics and engineering to model phenomena such as waves, orbits, and fields. 🌊

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Graph displaying polar function with $r$ and theta.

Image courtesy of Math Insight

When working with polar functions, it can be difficult to differentiate them using traditional Calculus techniques because the functions are defined in terms of r and θ, rather than x and y.

To illustrate, take a look at the graphs below:

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4 different polar functions.

Image courtesy of Lumen Learning

Although they are aesthetically pleasing, it sounds like a nightmare to actually differentiate them when looking back to our definition of differentiation in the Cartesian plane—the slope of the tangent line at a point—as this doesn’t translate well in Polar-ville.

To overcome this limitation, we can convert polar equations to Cartesian equations by using the following relations:

$$x=rcos\theta\\y=rsin\theta$$

Converting polar equations to Cartesian equations also allows us to visualize the functions more easily, as they can be graphed on a traditional x-y coordinate plane. This can be especially useful when working with complex functions that have multiple parts, such as a combination of trigonometric and polynomial functions. 🦄

Another conversion to be mindful is the following:

$$r=\sqrt{x^2+y^2}$$

#✏️ Converting Polar to Cartesian Practice

Let’s practice with some examples!

#🥇 Converting Polar to Cartesian Example 1

Convert the following polar function to a Cartesian function:

$$r=4\sin\theta$$

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Polar function r=4sin(theta) graphed.

Image courtesy of Sumi Vora

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