Trig Double Angle Formulas from Semicircle (visual proof)

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This is a short, animated visual proof of the Double angle identities for sine and cosine. To get the formulas we use a semicircle diagram and rely on similarity of two right triangles formed inside.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoff.... Thanks!
For another visual proof of this fact using the laws of sines and cosines, check out this animation:
• A proof to remember: D...
This animation is based on a visual proof by Roger B. Nelsen from the January 1989 issue of The College Mathematics Journal (www.jstor.org/... - page 51).
#mathshorts #mathvideo #math #trigonometry #lawofsines #lawofcosines #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #obtuseangle #acuteangle #angle #trigidentities #identity #righttriangle #isoscelestriangle #doubleangle
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Опубликовано:

14 окт 2024

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Комментарии : 34

@tacemus Месяц назад

Fantastic explanation. Really interesting. Thanks for all the effort you put in. 😊

@MathVisualProofs Месяц назад

Thanks!

@kindreddarkness 2 месяца назад

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...).

@MathVisualProofs 2 месяца назад

Thanks! Good luck with your math journey!

@llll-lk2mm 2 месяца назад

I love that idea, that's a very organic and structured way to go about it

@khoavo5758 2 месяца назад

I’ve attempted the same with NJW’s video and a theorem prover, sadly concluded that it wasn’t worth it…

@alanthayer8797 2 месяца назад

Thanks fada VISUAL Visuals visuals bcuz life AINT da same w/o them when seeking Precision through Teaching!

@MathVisualProofs 2 месяца назад

@nicepixels8076 2 месяца назад

I love the elegance of math I should say I once despised math, Now I love it and also score in it!

@MathVisualProofs 2 месяца назад

😀👍

@MichaelRothwell1 2 месяца назад

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.

@MathVisualProofs 2 месяца назад

Definitely could have used it. Just decided to justify it again for this case :)

@asepulven2768 2 месяца назад

I always wondered where that came from! Thanks

@MathVisualProofs 2 месяца назад

👍😀

@jaikumar848 2 месяца назад

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

@MathVisualProofs 2 месяца назад

Thanks! I've got some other trig in the queue, but running low on time these days. I'll get to it!

@theastuteangler 2 месяца назад

filled in a lot of blanks for me. thanks!

@MathVisualProofs 2 месяца назад

Glad it helped!

@llll-lk2mm 2 месяца назад

this is beautiful

@MathVisualProofs 2 месяца назад

😀👍

@EliazRK 2 месяца назад

First comment, epic Wow, you made it super easy to understand, my teacher used a lot of weird constructions to prove it 😮

@MathVisualProofs 2 месяца назад

Glad it helped!

@khoavo5758 2 месяца назад

Nice vid but I’m not sure if it’s a “visual” proof, or just a “visualized” one.

@GusBatista03 2 месяца назад

It's beautiful🥰

@MathVisualProofs 2 месяца назад

Thank you! 😊

@Dhanika112 2 месяца назад

Thank uuuuuu

@jakobthomsen1595 2 месяца назад

Very elegant!

@MathVisualProofs 2 месяца назад

👍😀

@Astro_Tutor 2 месяца назад

Very nice 👍

@MathVisualProofs 2 месяца назад

Thank you 👍

@HienNguyenHMN Месяц назад

Neat!

@jackkalver4644 2 месяца назад

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.