Does anyone know where the -2ab cosθ comes from in the distribute/multiply step? I don’t know how that could’ve been achieved by distributing and multiplying?

Great❤

Thank You For Nicely Explained...🫡🙏

You da man...

jeez ......simplicity at its best....such a life ssaver Thank You💯🙌

Hi, I really appreciate the effort of trying to proof e=c/a from the eccentricity definition of directrix. I had been trying to do the same and searched for the proof. Your video is the only one attempted this. I finally managed to proof it from geometry. The e must be the same at any point on the ellipse, so I equate the eccentricity at the two ends on the semi major axes, found an expression for d=(a^2/c)-c, where d is the distance from the primary focus to the directrix. Substitute that into an expression of e derived from the point at the semi minor axis, and that complete the proof.

Great video, thanks.

This applies only to perfect ellipses with to focal points multifocal ellipses have different shapes look at the orbits of saturn for example they have the same parameters but their shapes are different. Thanks for the video btw.

oh my goodness!! i was about to take my math final, and was worried about these questions, now hopefully i can get a better grade!! this was so insightfull, the way my homeschool taught me it, it did not really show much of a distinction, but now i get it way better, thank you so much.

please upload calculus series 🙏🙏🙏

This has long got me confused. Thank you for a great explanation

Your visualizations and comparisons to premutations really help to understand the concept, great video.

Your visual representations of objects really help, thanks for an explanation)

Hello again is there a way to define an ellipse based on the two foci with the property that a ray that starts at one of the focal points will reflect from the ellipse back to the other focal point? Thank you Arik

I am looking for good source to help me as a relatively new math teacher - I just found it - thank you from Israel

Good!🤜

*PERFECT*

Its CRAZY how someone on yt can explain a concept much more efficiently. I am a math major in the first semester and I am rlly struggeling to understand concepts when professors explain it, or its just hatd for me to understand stuff in the lecture, even worse when friends try to explain it to me… as they are trying to confidently teach me, (the themselves haven’t understood it good enough) I then feel very stupid. But I know it mostly depends on their explanation… so thank you!!!

From twelve years ago yet stands out neatly!

Very well explained👍

rare animation of equal distance to foci found!!

Thank You

Before finishing wathcing this video I didn't believe that this short 5 min video could actually help me but I was soooo wrong. Thank you so much.

Simple and to the point. Thanks for the explanation. My high school teacher came up with a method to never forget which is which and I'll share it just in case anybody is interested. Assign sine to your arm in a vertical position and cosine to your arm in a horizontal position.

Thanks a lot I didn't know how to get the other two equations from the original, so this really helped

This was best tutorial for understanding the p and c thank you

Thanks man

sort of really liked how it all clicks,when its contradicting the assumption itself.like in the first contradiction,you realize,the polygon area is smaller than my triangle,but the triangle is smaller than circle and the polygon area is clearly getting closer to circle so it must be greater than triangle,oh but that can't happen.I guess its wrong to assume that that the triangle area was less than the circle area in the first place.similarly for other contradiction.nice.I like your video.

Haha

Every other page in Google results is a piece of trash waste of time, thanks for your video, tho

very clear

Outstanding

Thanks for sharing! I posted a short video on deriving the law of cosines, It should apply for all angles (acute, obtuse, reflex, negative). Hope to get your thoughts.

Steve, nice presentation. What software are you using?

Great video. Thank you!

Thank you so much!

Euclidean method: three possibilities, prove two false then the remaining is the truth, e.g. in the comparison of two line lengths a and b there are three possible truths either a is greater than b or a is smaller than b or a is equal to b. Prove two of the possibilities false then the remainder is true. In Archimedes example he proves it false that the area of the triangle is greater than the circle and he proves it false that the area of the circle is greater than the triangle, therefore he is left with only possibility which must be the truth namely the area of the circle and the triangle are the same. Of course this exactness is dependent on the value of Pie and considering Pie is a transcendental and irrational number you can never get a perfect answer. Although a hand full of significant figures will take you to the moon, mars or beyond.

Great explanation!!!

Well described.

I bilieve that this explanation is wrong. 1/x is uniformly continuous if you wisely restrict the domain.

🤬🤬🤬

really nice proof ty

Thank you!

Thanks a million! 十分感谢！

thanks

At 2:16 you say "eventually as we keep on adding more sides to the polygon, it's going to become bigger than the circle." Can you please explain show how we know this is the case? For some reason, it isn't apparent to me.

Raj Sir ke fans kha ho?

Yes, there's a typo in the proof, as many commenters pointed out (thanks!). Hopefully you can see that it works out.

Beautiful explanation

what about tangent?