Hey, everyone. In this problem, we're asked to graph the equation r2=25⋅cos2θ. Now here, this is the equation for a lemniscate because I have this r2, which I know is only something that happens when dealing with equations of lemniscates. Now lemniscates are arguably our easiest shape that we know how to graph, so let's dive right in here. Now looking at our first step, we want to plot our first petal here, which is going to be located at a for our r value, and then we need to determine what θ is based on the trig function that we're dealing with. Now remember that in our equation, this value at the beginning is actually a2. So we need to make sure that we take the square root of that in order to get a. Now the square root of 25 is equal to 5, so this gives me my value of a that I will be plotting for that first petal. Now we need to determine what θ is. Now here I have 25 times the cosine of 2 θ. So I'm dealing with a positive a2 and a cosine. So that tells me that θ will be located at 0. So I can plot my very first petal at 5 0 which will end up being right here on my polar axis. Now all that's left to do is reflect this petal over the pole. So doing that here, reflecting this over to the other side, landing me right here. I have my 2nd petal, and I can move on to my final step and just connect these with a smooth and continuous curve. Now we know that the general shape of our lemniscate is a sort of infinity symbol or a propeller or whatever you want to think about this as, but remember that we are always going to go to that hole in there with these sorts of petals or propellers or whatever you want to call these. Now we've graphed this lemniscate at r2=25⋅cos2θ. Let me know if you have any questions.
Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
9. Polar Equations
Graphing Other Common Polar Equations
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