Visual Calculus: Derivative of sin(θ) is cos(θ)

Build an understanding behind different concepts of calculus that will help you tackle challenging problems at:
brilliant.org/ThinkTwice
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Proof: Derivative of sin(θ) is cos(θ)
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Support my animations on:
/ think_twice
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Any further questions:
Email: [e-mail]
Twitter: / thinktwice2580
Instagram: @thinktwice_ltu / thinktwice_. .
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Programs used:
- Cinema 4D
- Processing
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Music by Zeeky:
"Saw you"
• [[FREE]] Frank Ocean T...

[@JM-us3fr]

Dude, you’ll never run out of geometric proofs to do. These are awesome videos, keep up the great work.

[@xjonnyd93x]

Amazing animation ands good proof

[@eamon_concannon]

At 1:14 you did not explain why the angles labelled alpha are equal. The reason is that the upper alpha is the angle between the horizontal and the tangent. Since the radius is perpendicular to the tangent, we have alpha + theta = 90. The small triangle at top right is an approximation to the small dark region. The angle inside of it (indicated by alpha) is an approximation to alpha which approaches alpha as delta_theta approaches zero. This is a very nice proof and I love the channel.

[@Z7youtube]

thx for telling!

[@deepakjanardhanan7394]

It helped thank you 😊

[@Invalid571]

Excellent video/proof! 👏 👏 ☺ Edit: You deserve so much more subscribers, the quality of your videos is amazing! I shared this everywhere I can. ☺

[@ThinkTwiceLtu]

Thanks a lot for the support:)

[@alkamishra9974]

@@TheReligiousAtheists By your logic 3b1b's video wasn't original as well. He must have learnt this from somewhere its not like he created it.

[@Wild4lon]

The last part was a bit clunky. You didn't need to use similar triangles. Top left angle on the triangle is theta. Then it's obvious that cosθ = ∆y/∆θ just from using the smaller triangle.

[@ryanbell3704]

yo this is a genius comment

[@gasun1274]

never thought of it this way. the proof can be simplified into an infinitesimally small right triangle

[@kojo6492]

Why the top left angle is theta without using the similar triangle? I'm stucked here. Pls explain

[@i_am_wiz]

@@kojo6492 The top left angle is 90 - alpha….and since alpha + theta = 90, top left angle is equal to theta

[@alejrandom6592]

​@@kojo6492π/2+α+θ = π

[@stevemallot721]

Dude! Where were you when I was in high school? Oh, wait. Where was the internet?

[@GoogleUser-lz9nc]

we also had communists in our country then :(

[@dansman1729]

Typical american brainwashed by red scare propaganda

[@bmbiz]

@@dansman1729 Typical stupid comment by a yankee-hater.

[@josephcooper1928]

This video made me want to cry, it’s so eligantly beautiful. It makes me see why so many have pursued mathematics with the passion that they have. It truly is the pathway to hope and reason.

[@Magnasium038]

Beautifully done. Proof shown visually without any talking and minimal written explanation

[@michaelrosenberg6528]

SO THAT'S WHY WHY DIDN'T I THINK OF THAT? THAT EXPLAINS SO MUCH! GEOMETRY'S SO PRETTY THANK YOU!

[@fwcolb]

Yes, this is intuitive because the sine wave and cosine waves are 90 degrees out 0f phase.

[@bananaforscale1283]

Derivative of sin could be some function and derivative of cos could be this function 90 degrees out of phase.

[@Kokurorokuko]

intuitive doesn't mean it's proven

[@fwcolb]

@@Kokurorokuko True, but it's enough justification to do the work to prove the proposition. For some of us, time is of the essence. Problems that can be solved have higher priority than those with little chance of resolution. We leave those problems to others who value time less.

[@thekappachrist9540]

@@Kokurorokuko Just look at the graphs of sinx and cosx real quick. www.desmos.com/calculator/excenis4e4 Look at x = 0 where sinx has its largest slope. There, cosx has a value of 1 (the slope of sinx at 0). Now look at where sinx is at its highest value, at x = pi/2. The slope at the peak is zero, and we can see that cos(pi/2) = 0. The value of the cosine function clearly represents the slope of sine at any x value.

[@Kokurorokuko]

@@thekappachrist9540 it can be any other function with same characteristic points.

[@vpambs1pt]

Wow, powerful! Nice song as always, fits really well!

[@ThinkTwiceLtu]

Nuno Mateus thanks :)

[@jlpsinde]

First video from you I've seen and enjoyed very much, great video! Thanks our beautiful math never stops surprising...

[@JohnBicknell]

I can prove it analytical, but this is far more satisfying.

[@popcorn485]

The algorithm has served me well today. Now I have to go watch all your other videos!

[@XuanNguyen-op4qs]

Damn, the beat drops so sick, I can't even focus on the visualisation 😂

[@Zokalex]

I honestly didn't understand very well. I actually understand it more the mathematical way. I understand until 1:38, I don't know what the similarities are and what they play

[@mr2octavio]

Me too, that's when I lost the proof

[@ClancyXanecrest]

The similarity of the triangles comes from them extending the initial angle THETA by a tiny amount DELTA THETA. From here you can see they're similar because both triangles are RIGHT meaning ALPHA + THETA = 90° Then they show the series of step equations that demonstrate DERIVATIVE SIN(THETA) = COS(THETA) I recommend pausing at this point and stepping through the step equations yourself and compare each step with the triangles.

[@ClancyXanecrest]

@Bush Ninja You're correct, and perhaps I should've been more explicit on the condition in which the right triangle occurs but I don't think that's a reason to be rude as you were.

[@nikhilnegi9446]

You should first learn limit.

[@gentlemandude1]

Agreed. The explanation begins to fall apart at 1:38. I understand what's going on, but it would be very easy for someone who is less familiar with Calculus/Trigonometry to lose the thread of the explanation. It's a good video, but the lack of commentary makes it a bit opaque at some points. Watching it multiple times is useful.

[@svoidmist]

Sweet graphics as usual!

[@ThinkTwiceLtu]

Thanks:)

[@yaaryany]

Seen this before in one of the 3b1b's essence of calculus episode. And it's absolutely fantastically Awesome! You deserve way more subscribers!

[@mangalbaskey1825]

Presentation style is so good. Nice work.

[@atomiccompiler9495]

Thank you for uploading this today, I was having a hard day and this made a smile on my face.

[@xyz.ijk.]

Your work ... leaves me without words.

[@carladionotmorani8873]

Beautiful, thank you.

[@plominecraft8852]

Nice work man, keep it up.

[@OneWeirdDude]

That was awesome, you didn't even need trig identities. Good work.

[@Daniel-we7bx]

I love this kind of videos

[@jasonli1060]

never would've thought of this. brilliant

[@ToniBrasil100]

I never understood that way before....thanks a lot , from Brazil 🙏

[@iceiceisaac]

Awesome video, pls do more calculus! Maybe flip the triangles to match for the similar triangle bit?

[@AnonEMoose-mr8jm]

It's sad that math isn't being taught like this everywhere

[@gluonsmx]

Great animations! Do you code them or do you work in a software like after effects? Hope to see more of your stuff soon!

[@Name-is2bp]

hello think twice ,can you please make series and playlists on calculus and other math stuff that would help students..? thank you for the amazing videos!!

[@brucelickey748]

I'd love to see a follow-up of this demonstrating why the derivative of cos(x) is -sin(x).

[@davetheminion6653]

Bruce Lickey you could just say cos(x) = sin(pi/2-x) and differentiate this expression by using the chain rule. But a geometric proof would still be interesting :)

[@TechToppers]

This should be done as an exercise...? It's simple...

[@mehmetcanvurucu6836]

love your videos!

[@balajisriram6363]

such a beautiful proof!!!!

[@rodrigo-vl7bi]

This is so interesting, the algebraic proof is easy to understand but this one is even easier

[@no-qm1kn]

your channel is a gem

[@ThinkTwiceLtu]

ʕ ꈍᴥꈍʔ

[@connorcriss]

Congrats on the sponsorship!

[@funkysagancat3295]

Beautiful proof, thank you very much

[@shreyasnagare1286]

Keep up the good work.

[@BigDBrian]

note that the circle segment 'triangle' approaching a right triangle is essentially the limit as x goes to 0 of sin x / x being 1.

[@randomdude9135]

Seeing ur other videos, I thought by 'geometric proof', u meant a 'word less' proof. This is almost a formal proof. Idk.......... But, the animation was great, as always. Keep up the good work👊

[@zmaj12321]

For proving a derivative, this is about as wordless as you can get. I see what you mean though.

[@balaeinstein8710]

Thanks a lot for these beautiful videos . It is different way of approaching maths . It will be more helpful if you explain along with the animation .

[@deepaks.m.6709]

Awesome video! Thanks :)

[@prachisharma463]

Yoo! dope beats bro

[@Maxwell2010Macana]

outstanding! big thanks!

[@mikaeelshah7205]

Beautiful animation

[@msmmath87]

The best show! easy understand 👍

[@zukeow117]

Amazing video!

[@alvinlepik5265]

at 2:03 the third equality holds because the ratio is equal to x/1 as theta varies, thus the limit must also be x and the last equality holds because of Pythagorean thm. Good work

Really brillant!

[@abhinavchauhangujjar6456]

More of these please

[@manishkumarpradhan8176]

This video help me a lot.

[@BlueGrovyle]

I was surprised to see "Visual Calculus" in the title of a video not made by 3Blue1Brown. Guess I'm spoiled, but this video is pretty great too. Thank you for this!

[@rosishkatuwal5677]

Thank you very much for the video....

[@ytirucsbo]

I'm halfway through a semester of trig and I'm so lost lmao

[@PaulMarostica]

To Think Twice: I like the presentation style, but I think more careful visual explanation is necessary. I think you need to show +1s and -1s in the graph of the unit radius circle where it intersects the number lines, or it is not a unit radius circle. I think you need to show angles represented by nearby circular arcs. And I think you need to show at the beginning that all angles must be in radians only, because if they are not, then your derivation is not correct. In example, if all angles are in rotations, then I think arc length is 2 pi r delta theta, not r delta theta, and the derivative with respect to theta of sin(theta) is 2 pi cos(theta), not cos(theta). Any reply to my suggestions would be appreciated.

[@SUMITYADAV-mt8xn]

Okay I will mug up the formula. thanks

[@munib-munomactivity2442]

Need a lot of visual calculas video for better understanding

[@vijaybabu4082]

I subscribe your channel on a single video Because of this great presentation.

[@carollynbratcher3587]

Have you considered speeding up the info and maybe flashy coloring? I like what you got going on here. Have fun!

[@kaustubha7371]

Amazing

[@yafeiliu6969]

Thank you!

[@charitsfachrurizalkusumara5775]

Good job 🔥🔥

[@satyveer9865]

Awesome as always

[@ThinkTwiceLtu]

Thank you

[@satyveer9865]

@@ThinkTwiceLtu pleasure

[@anirudhj2134]

i cleared my maths paper after watching this one video. thanks ..!!!

[@xoxo-sf1zg]

Brilliant! 👍😎

[@engestruturas]

Excellent!

[@Singhamar313]

Thank you for this.

[@ThinkTwiceLtu]

:)

[@kirankumardhule]

Hey how u plotted this type of graph plz tell me software name🙏🙏🙏🙏🙏

[@MCatwar]

my physics teacher did something similar, showing how cos=-sin by graphing the waves on a plot and then graphing its acceleration

[@saadzaki1658]

Me after hearing the starting music, yea this is what i want

[@laxmipapney7182]

thanks bro, you are awesome......

[@SofiaYsabellesChannel]

Wow.. thank you for sharing this video. I admit I’m not good in Math, watching here from Australia

[@GrimmReaperRL]

I'm so dumb I was lost at the beginning so I didn't understand what was being proved lmao this isn't my strong suit (clearly) so I was hoping to learn a little more about this side of math. I'll just stick to basic algebra 😂

[@tffelpiano2830]

Why does the arc length equal the radius times delta of gamma?

[@mahmoudalbahar1641]

very great video because I love geometry and calculus very much and because the video is well prepared

[@ThinkTwiceLtu]

Thank you very much!

[@yimoawanardo]

Do geometric proof for the converging series that gives something with pi. The one where you multiply a buch of odd numbers!

[@zakirorrahman9007]

Well dude you did an amazing work absolutely amazed by the visualisation .Stopped playing League of Legends

[@ThinkTwiceLtu]

Thank you:)

[@Asv2609]

Wow great pls make more

[@mariorocha1005]

Gracias

[@anamarijavego6688]

I would like to recommend you an awesome book, Proofs without words 😉

[@OrangeC7]

Even Think Twice is sponsored by Brilliant, and they only have 50K subs? Well, I guess the sun never sets on the Brilliant empire... Wait that sounded like a compliment, no, that naming choice is Brilliant! NO THE PUNS ARE NEVER ENDING

[@e.matsuo9838]

Hermoso 😍😍

[@fromblonmenchaves6161]

why the smaller triangle has the angle alpha? Please explain.

[@mr.ricomathandphysicslover7257]

To understand d/d theta you must know what the definition of derivative itself that is by the concept of limit.

[@vijaysubramanian2037]

Do a collab with 3b1b!!... He did a similar/same proof for this in his video. But nonetheless keep up the good work!

[@Risulfur]

That's some death note deduction level shit. I'll take a sin derivative and PROVE IT

[@astro_mihir4485]

Amazing video

[@ThinkTwiceLtu]

thank you

[@farhanhassan1865]

This is good 👍

[@minhaj14d]

cool graphics. also make a video about how sinθ=tanθ can happen

[@JM-us3fr]

Isn’t that just when cosine is 1, which is every multiple of 2pi

[@gansach]

@@JM-us3fr then sine & tan would be 0,

[@JM-us3fr]

Gandharv Sachdeva I’m not saying it’s false. I’m saying it’s trivial

[@MrPorkered]

Wait I dont understand why does it become a straight line as delta titre approach zero? 0:53

[@mrshodz]

Great video. Can you please do a series on how Hindu mathematicians dervived sine, cos, arctan and pi and tan. What was their understanding of series. This would be a wonderful series😊.

[@sylvainm-v7919]

Clean

[@flavioryu5922]

So beautiful

[@rog3rskunk84]

someone explain me this at 0:53 what do u mean by approaching right angle triangle as \delta theta\ approaching to 0 aka the limit? if \delta theta\ = 0 then there would only be a straight line or what? its confusing me.. pls help

[@hakeemnaa]

the curve length will approach straight line length if the curve is very small as sin x ~= x if x is very small (Radian ) you can see it also like this half cycle curve=pi ,, line =2,,, difference=1.14 quarter cycle curve=pi/2,,, line square root 2,,, difference=0.16 the difference goes to zero quickly we divide the half cycle by 2 but the difference reduce by 1.14/0.16= 7 times so the difference is convergence to zero

[@darrenho3655]

Can someone explain to me a similar geometric method used for proving the derivative of cosθ = -sinθ ? Using the smaller similar triangle we can define ∆cosθ as the arc length increases (or labeled ∆x in this animation) to be the triangle side opposite from θ. From there I observe that sinθ = opposite/hypotenuse = ∆cosθ/∆x = d/dx(cosθ). Is there a logic step that I am missing? Where does negative come from? Please help!!

[@josephclaro4173]

Was thinking the same thing

[@darrenho3655]

@@josephclaro4173 I figured it out. If you have any questions lmk

[@alexanderbureyko4185]

@@darrenho3655 Could you explain, please?

[@SurinderKumar-os5il]

sir Plz. Define what is secant x. Tanx

[@manishamunde2220]

can anyone explain the part afte 1:26

[@cocacooler7]

How do you prove that as Δθ approaches zero, the dark colored regions becomes more and more like a triangle?

[@yashuppot3214]

The deriviative of a point on a circle is always the tangent line to the radius at that point.

[@PrashantKumar-hm8kn]

Sir make more videos about differential

[@sarojpandeya7883]

Thanks