In this lesson, our instructor Vincent Selhorst-Jones teaches the polar equations and functions. He discuses graphing in polar coordinates, tips for graphing, graphing equations of one variable, and converting coordinate types.
We can set up equations and functions using r and θ exactly the same way as we did for x and y. Generally, θ is the independent variable (like x was), while r is the dependent variable (like y was).
We always assume that θ is in radians ([(π)/2], [(7π)/4], etc.).
We graph polar equations/functions in the same way we graph "normal" stuff: plug in values, plot points, and connect with curves to make a graph. Since θ is the independent variable, we plug in some value for it, then see what distance r we get
out at that angle.
Just like graphing with rectangular equations, you don't need to plot a huge number of points-merely enough to sketch the graph. Many polar equations involve trigonometric functions. The "interesting" points are when the trig function produces a zero
or an extreme value (sin, cos ⇒ ±1). Figure out where these interesting values will occur and use them to help plot the graph.
If at anytime you're unsure how the graph will behave, just plot more points. The easiest way to work out your uncertainty is by calculating more points.
Occasionally you will see an equation that only uses one variable: that's fine! It means that variable is fixed, while the other can change freely.
Sometimes we'll want to convert an entire equation or function from polar to rectangular, or vice-versa. We can do so with the same conversion formulas we figured out and used in the previous lesson. Since these formulas were based off any x, y, r, θ,
they work fine for substituting in equations.
Polar ⇒ Rectangular: x = r cosθ y = rsinθ
Rectangular ⇒ Polar: r2 = x2 + y2 tanθ = y/x
If you have access to a graphing calculator, it's great to try graphing some polar equations with it. It's a new way of looking at graphing, so it helps just to play around. For more information, check out the appendix on graphing calculators.
Polar Equations & Functions
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.