You are an amazing teacher. I have started working through all of your videos with my trusty compass and straightedge. And I feel like I am finally actually learning math. Something that I have always wanted was a "Walk the Path"-course that derived modern high-school math entirely from classical findings/proofs/theorems, in order of historic appearance (i.e., this theorem made that theorem possible...).
Very nice! I think it's the first time I've seen a direct proof of the double angle formulae without using the compound angle formulae first. One question: why didn't you use the theorem that says the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference? Possibly because you can't rely on folks learning this at school any more.
I have a visual proof that d/dt sin t=cos t. Start with the point (1,1). Display “||dv/dt||=1”. Draw a circle by rotating that point around the origin. Display “||v||=1”. Combine both messages to say “||dv/dt||=||v||”. Then draw a tangent line at (1,0) pointing up and a vector from the origin to (1,0). Rotate the whole figure around the origin. Display “The angle doesn’t change.” Rotate the figure so the tangent line is vertical or horizontal. Display “The angle is 90°.” Merge that and “The angle doesn’t change,” to say “The angle is always 90°.” Replace that with “dv/dt=C*v.rot(90°)” Then display “||dv/dt||=C*||v||”. Merge it with “||dv/dt||=||v||” to say “C=1. Merge that with “dv/dt=C*v.rot(90°)” to get “dv/dt=v.rot(90°)”. Then change it to “d/dt =”. Change the right side of the equation to . Finally, split the equation into its components.
You are an amazing teacher. I have started working through all of your videos with my trusty compass and straightedge. And I feel like I am finally actually learning math.
Something that I have always wanted was a "Walk the Path"-course that derived modern high-school math entirely from classical findings/proofs/theorems, in order of historic appearance (i.e., this theorem made that theorem possible...).
Thanks! Good luck with your math journey!
I love that idea, that's a very organic and structured way to go about it
I’ve attempted the same with NJW’s video and a theorem prover, sadly concluded that it wasn’t worth it…
Fantastic explanation. Really interesting. Thanks for all the effort you put in. 😊
Thanks!
I love the elegance of math
I should say I once despised math, Now I love it and also score in it!
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Thanks fada VISUAL Visuals visuals bcuz life AINT da same w/o them when seeking Precision through Teaching!
:)
this is beautiful
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Really nice video !! will help to remember. Could you please make visual proof of cos A - cos B ? it coming a lot in my power electronics course
Thanks! I've got some other trig in the queue, but running low on time these days. I'll get to it!
It's beautiful🥰
Thank you! 😊
I always wondered where that came from! Thanks
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Very elegant!
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First comment, epic
Wow, you made it super easy to understand, my teacher used a lot of weird constructions to prove it 😮
Glad it helped!
filled in a lot of blanks for me. thanks!
Glad it helped!
Very nice 👍
Thank you 👍
Thank uuuuuu
Very nice! I think it's the first time I've seen a direct proof of the double angle formulae without using the compound angle formulae first.
One question: why didn't you use the theorem that says the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference? Possibly because you can't rely on folks learning this at school any more.
Definitely could have used it. Just decided to justify it again for this case :)
Neat!
Nice vid but I’m not sure if it’s a “visual” proof, or just a “visualized” one.
I have a visual proof that d/dt sin t=cos t.
Start with the point (1,1). Display “||dv/dt||=1”. Draw a circle by rotating that point around the origin. Display “||v||=1”. Combine both messages to say “||dv/dt||=||v||”. Then draw a tangent line at (1,0) pointing up and a vector from the origin to (1,0). Rotate the whole figure around the origin. Display “The angle doesn’t change.” Rotate the figure so the tangent line is vertical or horizontal. Display “The angle is 90°.” Merge that and “The angle doesn’t change,” to say “The angle is always 90°.” Replace that with “dv/dt=C*v.rot(90°)” Then display “||dv/dt||=C*||v||”. Merge it with “||dv/dt||=||v||” to say “C=1. Merge that with “dv/dt=C*v.rot(90°)” to get “dv/dt=v.rot(90°)”. Then change it to “d/dt =”. Change the right side of the equation to . Finally, split the equation into its components.
"why so underated?"
?
Gg
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