Graphing Polar Equations | Calculus 2 Lesson 47 - JK Math

[@JKMath]

︎►►📝Download my free blank polar coordinate systems to help you practice graphing: www.jkmathematics.com/blank-polar-coordinates-worksheet 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!

[@whatever9537]

thankyouuu, u're a life saver

[@thembastaff]

Auto corrected that 😅 thanks for the clarity though

[@sallywang7983]

Hi Sir, I am slightly confused with the last two graphs. You said to rotate 180 degrees, but the graph you are drawing looks like you rotate 90 degrees, not 180 degrees. Am I wrong? Thanks for your reply.

[@JKMath]

@@sallywang7983 Hi! Did you read my full comment above? Under minor corrections I mention that i did misspeak slightly at 46:21. I don't want to repeat myself, so I'd recommend you read my correction and then if you still have any more questions please let me know!

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[@marymoser5438]

Thanks!

[@JKMath]

You're welcome! And thank you for your generosity!

[@marymoser5438]

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.

[@JKMath]

Awesome! Thanks for sharing, glad the videos were able to help. Precalculus is definitely in my roadmap for future series to make one day!

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you deserve more subscribers.

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I have to share this video to my lecture, maybe then he'll know how to tech his students a new topic correctly

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The most underrated chanel

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Highly underrated video .shame on KZbin

[@davidvilla1247]

I have an exam on Wednesday 10/30 and wow you explained this so good, much better than OCT. I liked how you included absolute value of A-B and saying how to graph petals with sin and cos.

[@JKMath]

Glad I could help! Best wishes on your exam, you got this 👊

[@davidvilla1247]

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[@JKMath]

Thank you! I appreciate the kind feedback :)

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[@JKMath]

Thank you, I appreciate the kind feedback :) Feel free to share these videos with others!

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[@JKMath]

Thank you! Glad to hear the video was helpful :)

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Thank you so much for this explanation, I can't tell you how much help this was!

[@JKMath]

You're very welcome!

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That was a great explanation, thank you soooo much❤

[@JKMath]

You're very welcome!

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Thank you! You saved me from my reporting!

[@JKMath]

Glad I could help! :)

[@urboi_1238]

Thank you very much you are saving my precalc grade in this new trimester bro. You are an awesome teacher🙏👍😎

[@JKMath]

You're welcome! Glad the videos are helpful for you :) In the future if you take Calc 1 after precalc, be sure to check out my Calc 1 playlist as well!

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You are absolutely amazing. I will be using your videos from now on. Can you please tell me what program you use to digitally write on? I use explain everything but it doesn't automatically change my lines to perfect lines or my circles to perfect circles.

[@JKMath]

Hi! I use an app called "Goodnotes" on an iPad. The automatic shape drawing feature is definitely super nice!

[@Adam.Ab9898]

Great Video

[@pizzaman9859]

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.

[@JKMath]

You're very welcome! Glad the video was able to help :)

[@i7zu5]

THE GOAT!!!!

[@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?

[@JKMath]

Good questions! Yes, it may seem odd that we could have r² equal to a function with a negative coefficient, however in this case there are no imaginary/complex numbers to worry about. The reason is because sine and cosine have some nice properties. Specifically, we know that -sin(θ)=sin(θ+π) and -cos(θ)=cos(θ+π). So in both cases for the "infinity-shaped" lemniscates, if you have a negative coefficient, you can remove it by adding π to the angle. This is what I meant by "rotating by 180° or π radians." I misspoke slightly, and I apologize for that: yes, in terms of the actual graph you are really only rotating 90° or π/2 radians, but what I meant/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. Hope this helps!

[@Sam-the-chosen-one]

@@JKMath Thanks for replying this fast. It sure did clarify both of my doubts.. tomorrow we have our mid sem examination thanks a lot.. now I understand that pi was dissolved in 2(theta+pi/2).

[@JKMath]

@@Sam-the-chosen-one No problem! Glad I could help. Best wishes on your exam! :)

[@mathswithsksharma]

Very brilliantly explained, thanks a lot

[@JKMath]

You're welcome!

[@thembastaff]

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[@JKMath]

You're welcome! Glad the video could help :)

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[@JKMath]

Thank you! Glad the video could help you!

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[@zhongli7468]

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

[@JKMath]

You’re welcome! And yes, I mention the small mistake in my pinned comment on this video. I explain what correction should be made there.

[@daniellarson887]

Thanks for the refresher on precalc lol. Even did r value analysis. you are the best

[@JKMath]

You're welcome! Glad to help :)

[@Eren-me8mt]

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[@Rikudabest]

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[@VishalReddy.k-p3j]

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[@BrianCruz-tx7ff]

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?

[@JKMath]

For the quick method, we really only look at where the graph would cross the vertical and polar axis. If you start looking at other angles, then it really isn't so "quick" anymore. You would be better off using the plotting points and symmetry method if you want to be that accurate. That being said, typically, you want to know the general shape of the different polar graphs, and that should help you draw them quickly in conjunction with the points on the polar and vertical axis for the quick method. If you really wanted to, you could plug in the angle, such as π/4, into the polar equation you are working with, and see what value of r is outputted, and then use that to help you determine where the graph crosses that angle in the coordinate system. But once again, at that point you might as well just plot points and use symmetry. The way I see it, the quick method is really just for sketching the curves super fast, which helps later on in Calc 2 such as when you need to draw a picture to help you figure out how to set up integrals to calculate area between polar curves. If you need to be more accurate with the graph, you want to plot points and use symmetry instead. Hope this helps!

[@sevval9999]

Thank you so much. Great video

[@prathanapatel.9034]

I’m dumbest in math but you made it so easy Thankyou so much 😭❤️

[@JKMath]

You're welcome! Glad to help :)

[@ENS.-wr3eb]

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

[@JKMath]

For the plotting points and using symmetry method, you can pick whatever angles of theta you want to create some points. It is totally up to you. Now, you probably want to pick angles that are nice to evaluate for the given equation, so I recommend sticking with the most common angles of 0, π/3, π/4, π/6, π/2, π, 2π, etc. Those are usually the easiest to work with. Sometimes you can pick a weird angle like π/8 if that makes it easier to plug in, such as for the function r=cos(2θ). Because θ is multiplied by 2, using π/8 becomes convenient since 2 times π/8 is π/4, which is a nice angle to evaluate cosine at. Does that make sense? In general, just pick the most convenient angles. Hope this helps!

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[@Hailey94960]

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[@SkySouthern]

cos(A-B) = cosAcosB + (not -) sinAsinB at minute 17. luckily your sinA comes out to zero so...

[@JKMath]

Yep, I mentioned this error in my pinned comment. My apologies on that mistake!

[@zianiera]

Thank you very much