In this video I further into Polar Coordinates and this time look at Example 10 which involves graphing the function r = sin(8ϴ/5) using the amazing Desmos graphing calculator. Similar to my last video on graphing polar curves, I show how we can use the Desmos calculator to graph polar curves directly or by converting them to Parametric curves. Also in this video I show to determine the domain of the angle ϴ in which the polar curve repeats itself. I do this both visually using the graphing calculator, as well as analytically by using the periodic rotations of a typical sine wave. Note that this is just a quick and simple illustration of determining the domain for the simple case that the function does in fact have an exact whole number of rotations before it starts repeating itself. The curve graphed in this video is an amazing circular spiral flower shape and shows just how amazing the types of curves that can be plotted in polar coordinates (or as parametric equations)! Try it out yourself and let me know of any crazy shapes you can create!!

You can use my graph as a starting point: https://www.desmos.com/calculator/raiwkuult1

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Polar Coordinates: Example 10: Graphing r = sin(8ϴ/5)

Example

Graph the curve r = sin (8θ/5).

Solution:

https://www.desmos.com/calculator/raiwkuult1

Note, we can graph the same curve by also converting the polar curve to Parametric Equations:

To determine, analytically, the domain of θ, we can ask the question: How many complete rotations are required until the curve starts to repeat itself?

Recall that one rotation is just 2π radians, or 360 degrees.

If the answer is n rotations, then we must have: