Trig Double Angle Formulas from Semicircle (visual proof)

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.

For another visual proof of this fact using the laws of sines and cosines, check out this animation:

#mathshorts #mathvideo #math #trigonometry #lawofsines #lawofcosines #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #obtuseangle #acuteangle #angle #trigidentities #identity #righttriangle #isoscelestriangle #doubleangle

To learn more about animating with manim, check out:

[kindreddarkness]

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...).

[tacemus]

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

[Blob__Ship]

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

[alanthayer]

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

[asepulven]

I always wondered where that came from! Thanks

[jaikumar]

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

[theastuteangler]

filled in a lot of blanks for me. thanks!

[EliazRK]

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

[MichaelRothwell]

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.

[khoavo]

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

[jackkalver]

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 <cos t, sin t>=<cos(t+90°), sin(t+90°)>”. Change the right side of the equation to <-sin t, cos t>. Finally, split the equation into its components.