Graphing Polar Equations | Calculus 2 Lesson 47 - JK Math

How to Graph Polar Equations (Calculus 2 Lesson 47)

In this video we learn how to graph polar equations. This includes polar equations that represent basic polar graphs such as lines and circles, as well as more complex and special polar graphs such as limacons with an inner loop, cardioids, limacons with a dimple, convex limacons, rose curves, and lemniscates. We look at a method that will allow us to graph any polar equation by plotting points and making use of symmetry, and then we look at rules to help us quickly sketch special polar graphs by recognizing the form of a polar equation.

⬇️ 📝 Download My Free Blank Polar Graphs Here:

This video series is designed to help students understand the concepts of Calculus 2 at a grounded level. No long, boring, and unnecessary explanations, just what you need to know at a reasonable and digestible pace, with the goal of each video being shorter than the average school lecture!

Calculus 2 requires a solid understanding of calculus 1, precalculus, and algebra concepts and techniques. This includes limits, differentiation, basic integration, factoring, equation manipulation, trigonometric functions, logarithms, graphing, and much more. If you are not familiar with these prerequisite topics, be sure to learn them first!

Video Chapters:
0:00 What are Polar Graphs? (Intro)
2:10 Graphs of r=a & θ=a
4:50 Graphs of r=a*secθ & r=a*cscθ
8:19 Graphs of r=a*cosθ & r=a*sinθ
10:44 Graphing Using Points & Symmetry
12:37 Example: Graphing r=1+cosθ (Points & Symmetry)
26:08 Limacons With Inner Loop
30:36 Cardioids
32:25 Limacons With Dimple
34:33 Convex Limacons
37:11 Rose Curves (cosθ)
40:34 Rose Curves (sinθ)
44:58 Lemniscates
47:56 Outro

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-Josh from JK Math

#calculus

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[JKMath]

︎►►📝Download my free blank polar coordinate systems to help you practice graphing:

Also, MINOR CORRECTIONS: At 16:17 when I give the subtraction identity for cosine, I made a slight error. It should be cosAcosB + sinAsinB. I wrote a minus sign in between the terms by accident and did not catch it in the editing process since the final answer was not effected by the mistake (adding/subtracting 0 produces the same result!). I got lucky, but that will not always be the case when checking symmetry! Be sure to remember that it is (+) not (-). My apologies on that mistake!

At 46:21 I say to rotate the lemniscates by 180° or π radians when the coefficient is negative, and I misspoke slightly. For the actual graphs you really only rotate by 90° or π/2 radians (just like I show visually), but what I should have said is that each angle θ you use in the equation should be increased by π (or 180°) to adjust for the negative coefficient. This is because sine and cosine have the properties of -sin(θ)=sin(θ+π) and -cos(θ)=cos(θ+π). This is important because then if you were to solve for r by taking the square root of both sides of the polar equation, you don't have to take the square root of a negative number. The negative coefficient can be removed by adding π to the angle. Once again, I apologize for any confusion!

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[TwiningLion]

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[LJS-sznm]

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[Darkwater-swww]

Great video. It is a shame that it is so underrated.

[marymoser]

Great video, polar was included in the new AP Precalculus and I needed a good refresh to be ready to teach students when I haven't done anything with polar in decades and your videos were clear and helpful. And y If you ever get to a precalculus series I will include you on my recommended resource list for students.

[thembastaff]

This video was very very helpful 🙂 much appreciated 👍

[Sam-the-chosen-one]

great explanation just one doubt at 46:21 1) if the coeffetient a sqyared turns negative won't the value of r be imaginary/complex
2)rotatibg the graph by 180 degrees won't make any change on them right.. why did you create the graph rotated by an angle of 90 degrees?

[ndumisothulani]

Yor lessons are very helpful sir.thank you very much i really i appreciate your videos❤️🙏

[ENS.-wreb]

How do we know what thetha we will plug in the equation?

[movieshortclips]

Really this is a very underrated channel. Very informative video indeed

[BrianCruz-txff]

How do you know to where do the cardioid/limacons’ points extend for the other angles? Like how do I know if it crosses or not trogh pi/4 for example, after sketching the graph with the quick method?

[zhongli]

Thanks for such a good video. Just to point out a mistake, at 16:18 the compound angle formula is wrong.

[pizzaman]

Thank you for this video this helped me so much I was so confused on how to graph the thing and my previous difficulties with the unit circle only made it worse. Thank you so much for the good info and easily understandable content.

[PhilaniMngomezulu-im]

This video helped me a lot. You have earned my like and subscription :)