All 6 Trig Functions on the Unit Circle

Computer animation by Jason Schattman that shows how sine, cosine, tangent, cotangent, secant & cosecant all fit together in one beautifully unified structure on the unit circle.

Along the way, you'll see animated visual "proofs" of the 3 classic Pythagorean trig identities

sin^2 + cos^2 = 1
tan^2 + 1 = sec^2
cot^2 + 1 = csc^2

plus a 4th one I'm 99% certain you've never seen! In fact, I only learned it myself while creating these animations!

As the grand finale, you'll see how a spinning wheel creates the wave-like graphs of sine & cosine, and also the vertically asymptotic graphs of tangent, secant and cosecant.

I coded these animations using the Processing programming language, and annotated them using EquatIO.

More math animations from my channel
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On the beautiful geometry of imaginary numbers & complex functions (can be enjoyed without knowing what that means)

Mathematical art using the idea of epicycles

Optical illusions made using trigonometric functions

Fancy "card tricks" animated using mathematical pretzels (called Lissajous curves)

Sound waves in an oval room:

Fly through the 3D Sierpinski pyramid:

Drawing on a spinning white board:

[papahemmy]

I've been learning and using trig for 6 years now and this is the first time I've seen an intuitive example of all six trig functions acting together.

[WEPayne]

50 yrs ago I learn this from black & white drawings in a textbook. As I struggled to master it in my mind I would try to animate the dry motionless paper drawings.

Now you have brought to life so beautifully what I tried to imagine years ago it brings a tear to my eyes.

THANK YOU !!!

[camerongray]

They should have showed us this in school. I am good at math, but this visual would have made it soooo much easier to learn

[muqtarjamaegal]

This is most beautifull math animation ever thank you so much for your dedication

[_goldfish]

For about 2 years, I’ve been looking for an actual demonstration as to what the sin, cos, and tan functions ACTUALLY do, and I never got an actual answer. Then some random video in my recommended gives the PERFECT answer to my 2 year question.

THANK YOU

[irishamerican]

I like the way you arranged the triangle at 4:09 I struggled to understand what tan was, but the day I realised it was the slope was awesome & this arrangement shows tangent in its true form. Amazing video.

[giorgiocanal]

The visual association of functions and COfunctions to angle and COmplementary angle is simply beautiful.
Great job: this is the way math should be taught.

[YoshisaurUnderscore]

This is legitimately the coolest and cleanest visualization of the trig functions I have ever seen. I'm currently halfway through an engineering bachelor's degree (so 6 years of dealing with trig functions) and I still feel like I just understood trigonometry in a whole new light. Amazing animation!!

[Bananabeacon]

I love this so much! It's so intuitive. It really shows how all of these 'functions' are not just made up by someone, but rather how they have been Found and assigned their names! Like how the tangent is actually tangent to the circle, or how the secant (which, as you said in another comment, means "to cut" from Latin) actually cuts through the circle! All wrapped up in a nice and clear animation.

And then the music was just so cool! Not distracting, fitting and just great. It reminds me of those old videos from when people were first experimenting with electronic sounds.

Really well done!

[Jsuarez]

This is beautiful. I wish my math teachers in high school and college would have showed me this.

[simpleman]

I feel like I'm about halfway to understanding the triangle. Each time I understand one more small piece I feel like I'm floating among the clouds.
I understand enough to say this is an amazing video, I LOVE it.
I put it on loop & turn up the volume.
I had to stop it at 5:50 when I saw cot & tan were =. I had to work out each one: (Sin & Cos = 0.707 )(Tan & Cot = 1)(Sec & Csc = 1.414)
Without a doubt this is Beautiful Math. I know this video is your baby, but I'm claiming it too. Thank You so much for sharing it with us.

[TheEldad]

You know it is a good math animation when you hear psychedelic music in the background

[dereklenzen]

The screenshot at 4:15 is absolute gold. Rarely have I seen such a concise and meaningful representation of a mathematical concept. Well done!

[ItsPouring]

I'm so glad I clicked on this video suggestion. Math seems more like a life-long study than something you just do in school.

[BrokenLifeCycle]

I've learned more about trigonometry in this video than I have in every year of formal education that I've had in my entire life.

[GTAdkdk]

My mind is blown after seeing tangent line ACTUALLY being the tangent line omg. And how all the lines are organized suddenly makes so much sense. This very explanation should be done when trigonometry is first taught to students. Now I'm equipped with this strong intuition, all algebraic expression makes sense as well. I'm now taking on trig integration techniques with much more ease. Hats off to you and thank you!!!

[duckyoutube]

Its easy to imagine this the more familiar you become with a right triangle. But to see it in a video make the magic so much more clearer.

Mathematics is nature. Its the language of the trees, of the planets, of lightning, of music. Math is everywhere nature is.

[iamdino]

Very enlightening video. I'd never seen the triangles arranged in that way, making it evident why each function has their "co-" counterpart. Thank you

[surVERXD]

Wow never clearly understood trigonometric concepts, I just blindly learned the values, identities, formulas etc. Everything became beautifull....
Thanks for this beautiful math video!

[THEDeathWizard]

This video would probably be so helpful to kids in school learning this for the first time, especially if they’re able to interact with the diagrams and adjust the values until it feels intuitive for them. A lot of math teachers simply aren’t equipped to explain this using just a chalkboard, so hopefully animations like this will get more and more accessible as time goes on