All 6 Trig Functions on the Unit Circle

[@papahemmy8587]

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.

[@beautifulmath5361]

That is fantastic to hear! Thank you!

[@darrennew8211]

@@beautifulmath5361 How much easier would high school have been for me if they could produce something like this back then. How awesome.

[@andrewsemenenko8826]

​​@@darrennew8211 btw thanks for the "triangle form" visualized 5:27 (csc^2+sec^2=(cot+tan)^2). I think this one is the best here because it feels complete and consists of just one additional line (the orthogonal one). Also it is the least crowded representation, where every line has its separate place!

[@beautifulmath5361]

@@andrewsemenenko8826 Excellent point! 🙂

[@tizurl]

my math teacher made sure we used the trigonometric circumference for everything trigonometry related so even if we forgot relations between angles we’d know how to get them. also used them for demonstrations fairly often, i really appreciate it

[@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 !!!

[@beautifulmath5361]

What a wonderful story! I'm delighted this video helped you in this way. I too saw a textbook drawing of this setup, and I always wondered how it would look with different values of theta. Now in the age of computer animation, we can bring such diagrams to life!

[@alberttanner408]

Your 66 years of age. I assume if you were reading that text book at 16.

[@muqtarjamaegal6071]

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

[@scottl.1568]

Music needs work, though

[@redoni3429]

Can recommended the Mandelbrot set animated

[@beautifulmath5361]

@@scottl.1568 It's a free track provided by KZbin. ;-) It was handy and fits the ethereal mood of the math.

[@Z7youtube]

@@beautifulmath5361 i really like it and it fits perfectly with the video!

[@beautifulmath5361]

@@redoni3429 Do you have the link? I'd love to watch it.

[@camerongray7767]

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

[@beautifulmath5361]

I agree! I wish I'd seen this myself when I was in school. I made it precisely for people like me. 🙂

[@cmyk8964]

I think I _did_ see this kind of visual in my textbook, just not animated.

[@DadicekCz]

​@@cmyk8964 same for me, thank god

[@j.d.snyder4466]

I would've given just about anything to have had this superb clip many decades ago. Trig all but did me in back then.

[@beautifulmath5361]

@@j.d.snyder4466 I would have too! (Graduated 1991). I made it for exactly this purpose.

[@_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

[@beautifulmath5361]

I'm delighted this video was helpful to you! This interpretation of sin θ and cos θ is crucial to classical physics and engineering.

[@giorgiocanal1659]

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.

[@beautifulmath5361]

Thank you!

[@Quroe_]

Is that what the CO means?!

[@novarender_]

​@@Quroe_ 5:50

[@Max-ys1dw]

​@@Quroe_ yep. Seems like we could have been told that the first time we learned about cos, csc, and can doesn't it. Gotta wonder why we weren't.

[@user-xy5yg6se1k]

math is beautiful

[@irishamerican4558]

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.

[@beautifulmath5361]

Yes, that's where the word "tangent" gets its name. Likewise "secant" comes from the Latin "secare", which means "to cut". Thus, the secant line cuts across the circle and through it.

[@Blaster_Unity_UB]

@@beautifulmath5361 that's really cool

[@user-pr6ed3ri2k]

tan(a)x is a linear graph but it rotates smoothly unlike ax so I guess this has something to do with that

[@NickWrightDataYT]

I agree, that's much more of an intuitive arrangement of the values. Though I understand the first arrangement, since it's best suited for drawing those graphs.

[@erutuon]

I've seen diagrams of the trigonometric functions on the unit circle many times before, but this part of the video had the first diagram that made it clear to me why half of them are "co" versions. Thank you to @Beautiful Math. That massively helps with my uncertainty over which one is sine and which is cosine. I kind of know but now I have a diagram I can put in my head to be sure.

[@duckyoutube6318]

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.

[@simpleman283]

For sure, there could be nothing without math.

[@beautifulmath5361]

Beautifully said! I agree wholeheartedly!

[@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!!

[@beautifulmath5361]

Thank you for your kind words! I am delighted this animation was helpful for you. 🙂

[@mosantw2014]

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!!!

[@beautifulmath5361]

Thank you for your kind words! I'm glad this video was helpful for you!

[@Jsuarez6]

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

[@beautifulmath5361]

Thank you! I wish the same for when I was in school myself. It's what inspired me to make this.

[@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!

[@beautifulmath5361]

Thank you! So much of "hard" math really is intuitive if it's taught from a visual perspective.

[@kyledavidson8712]

Great synths I love the correlation math has with music

[@beautifulmath5361]

@@kyledavidson8712 It's just a free track provided by KZbin, but the ethereal feel of it fits the mood the mathematics.

[@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.

[@beautifulmath5361]

Definitely! Most of the math I know I learned after finishing school.

[@simpleman283]

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.

[@beautifulmath5361]

Thank you, Simple Man! That means a lot. My raison d'etre as a teacher is to highlight how the logic of math and the beauty of math fit together, like a hand in a glove.

[@skilz8098]

Another thing about the geometry of the triangles that you don't see being taught too often is that there is a direct correlation of an area of a triangle in conjunction with the trig functions. Consider the right triangle in standard form on the unit circle and let's say the the hypotenuse of the triangle has a linear slope of 1. This creates a PI/4 or 45 degree angle that is both above and below the line y = x. The point on the circle we know them as (cos(t), sin(t)) where x = cos(t) is the distance in x, and y or f(x) = sin(t) the height or the distance in y or f(x). Here the area of the right triangle that is generated by the origin (0,0), the point (x,y) on the circle and the vertical perpendicular bisector at x is 1/2 *xy since x is the base and y is the height. And here we know that x is cos(t) and y is sin(t) so the area of the triangle can also be written as A = (1/2)cos(t)*sin(t). When theta = 0. The hypotenuse will equal 1, the base will equal 1 and the height will equal 0. Here you have two lines that became parallel that are also also coincidental as there is no angle or distance between them. They are also coincidental with the x-axis. Here the slope or tan(t) is 0. We can see this from (1/2)(cos(0)*sin(0)) = (1/2)(1)*(0) = 0 for the area of the triangle and we can see this from the slope of the line sin(0)/cos(0) = 0/1 = 0. You now have a triangle with 0 area. Now since tan(0) = sin(t)/cos(t). The tangent exist when the area of a triangle is 0 since sin(0)/cos(0) = 0/1 = 0. When sin(t) becomes 0, y or f(x) becomes 0. We can see this from the point on the unit circle at (1,0). Now when the inverse happens and x becomes 0 and y becomes 1 on the unit circle when the point is (0,1). Here, we end up with a vertical slope since sin(90)/cos(90) or sin(pi/2)/(cos(pi/2) = 1/0. Here the right triangle that was under the hypotenuse which always has a length of 1, it's base in x or cos(t) is now 0, and the height y or sin(t) is now 1, the hypotenuse and the height or sin(t) have now become coincidental with the y-axis and are perpendicular to x or cos(t) and are parallel with each other. This then gives you a series of triangles where their areas are approaching infinity but instantly snaps to 0 once sin(t) becomes 1 and cos(t) becomes 0. Here we have vertical slope as in sin(t)/cos(t) = sin(90)/cos(90) = 1/0. Division by 0 and here the tangent is considered undefined because of division by 0. However, I like to think of it as approaching infinity and is ambiguous, because any slighter value greater than 90, the signs of some of the trig functions change. This change in sign I think is related to the even and oddness of the functions... These are wave functions and the sine and cosine are 90 degree translations of each other. So there is a phase shift that is happening. The area of the triangle is approaching infinity but never reaches it and then goes to 0 when sin(t) = 1 and the hypotenuse becomes vertical. Then when theta becomes greater than 90, the sine is still positive in the second quadrant but the cosine becomes negative and so does the tangent. Here the triangle is now reflected to the left side of the circle and the area instantly goes from 0 to approaching negative infinity since the hypotenuse is no longer coincidental with the y-axis and is no longer vertical but is now reflected past the y-axis. It is these properties of the triangle that define the inscribed properties of numbers and other functions that are based on reflective properties and symmetry. The behavior of what is seen within the area of the triangle is also proportional to sin(t), cos(t) and tan(t). The other 3 trig functions are just their reciprocals. When tan(t) = 0, the area of the triangle = 0. when tan(t) = und or 1/0... the area is also 0 at that point but was either approaching or coming from +/- infinity. This approach to an infinite area but never getting there is when the hypotenuse and sin(t) coincide and this is where the vertical asymptotes within the tangent function show up... I know this isn't quite as elegant as a video. But I find these patterns and connections to be intriguing to help better understand why numbers and functions behave in the way they do. When you look at the equation y = mx+b where m is the slope of the line defines by (y2-y1)/(x2-x1) = dy/dx we can see that dy = sin(t) and dx = cos(t). And this relationship of the slope of a line m is the same thing as tan(t) where the angle t is between the line y=mx+b when b = 0, and the x-axis. And since dy=sin(t) and dx=cos(t). We can clearly see that dy/dx = tan(t). And this gives us the foundation into derivatives and integrals. Algebra, Geometry, and Trigonometry are all related and are basically the same thing but represented differently... And what's even greater about the properties of triangles and the trig functions that they produce is that the trig functions are wave functions and we use them in physics, chemistry and other sciences to map energies such as sound and light, to map wave patterns, things that rotate, oscillate, vibrate or resonate, etc... The trig functions are wave, circular, oscillatory, periodic, and transcendental functions. Being able to relate the area of a triangle to that triangle's corresponding trig functions is another way to look at their properties and behaviors as a whole. This can help to give greater meaning when you start using these mathematical functions within the sciences such as in physics and chemistry. Now you can better understand the wave functions and what's happening within things like Schrodinger's Equation... Just food for thought...

[@beautifulmath5361]

@@skilz8098 Wow, thank you for this thorough analysis. I'd never thought about measuring the area of the triangle as the tip rotates!

[@skilz8098]

@@beautifulmath5361 If you think that was something... try this one on for size... the very first or simplest of all arithmetic calculations 1+1=2 is the basis for the unit circle except that it isn't located at the origin (0,0). This unit circle has its center located at (1,0). And if we plug this into the Pythagorean Theorem A^2 + B^2 = C^2 well, how can we? There's no right triangle here. We do have two unit vectors that lie on the x-axis V0 = P1(1,0) - P0(0,0) and V1 = P2(2,0) - P1(1,0). These two vectors are on the same line, so their angle between them is 180 degrees or PI radians. Let's take V1 and rotate it about the point (1,0) heading towards the y-axis, so that its head at (2,0) inscribes an arc. When we have rotated this by 90 degrees or PI/2 radians in a CCW direction. We have a right triangle with two sides that have a length of 1 and a hypotenuse with a length of sqrt(2).. 1^2 + 1^2 = C^2 = 2 = C^2 = sqrt(2). This can also be used to show a proof that the equation of the circle (x-h)^2 + (y-k)^2 = r^2 is just a specialized form of the Pythagorean Theorem. Hmm? An equation that defines a circle is a special case or derived version of a Theorem that is based on the properties / ratios of the length or magnitude of the legs of Right Triangles... And this can also show that even your radicals such sqrt(2) are imbedded in basic arithmetic as seen from above in 1+1= 2. At first glance when you look at that simple arithmetic equation, you'd never think of a Unit Circle, the Pythagorean Theorem, Radicals, but yet it's all embedded in basic arithmetic, it's all embedded within the ability to enumerate or to count. It's little things like these that isn't commonly taught that you end up picking up on your own that makes math and numbers so intriguing...

[@erinmcdonald7781]

@@beautifulmath5361 Wow. Thanks to both you and @skilz8098 for sharing your insights and elaborations on the humble unit circle. Even before watching your video, I intuitively sensed that something like this was true, but didn't have the ability to express it. Having this video to show students I tutor is going to help immensely. As for the additional breakdowns provided in the comments, you're right. This is why we study math because it explains and connects the concrete and abstract in elegant mind-blowing simplicity. Again, I had come to some of those conclusions, but was unsure because I hadn't seen them presented in that way until now. Thank you! 💜🌎📐🍀

[@iamdino0]

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

[@beautifulmath5361]

Thank you for this kind feedback! I'm delighted the video was helpful for you.

[@don.3s]

Having these kinds of videos back in school would be a blessing.

[@beautifulmath5361]

I agree! I wish I'd seen these relationships while in school, too!

[@TheEldad669]

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

[@buddydog1956]

you musn't have lived during the 60's ....that's not Psychedelic music....it's at best, 'electronic music' ~

[@TheEldad669]

@@buddydog1956 שכנעת אותי

[@1nfinitezer0]

the choice of epic synth music is very nice, and not distracting like the large majority of music used in videos

[@beautifulmath5361]

I'm glad you like it! It's just a free track I picked from KZbin's menu, but it happens to fit the vibe of the math I'm trying to show very well. And I agree about other videos. Most background tracks are just garish and distracting.

[@memirandawong]

I've had high school and college trig classes and NO ONE explained these identities better than this video, and I never heard a word!

[@beautifulmath5361]

Thank you! I'm so glad it was helpful to 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!

[@beautifulmath5361]

You are so welcome! I am glad this video was so helpful to you!

[@X_INTENT]

Man.. if only I found this video sooner. How didn't anyone explained me trigonometry like this

[@beautifulmath5361]

Thank you! I'm so glad this was helpful for you!

[@Name-xd1hv]

Although I'm 13 and still learning rather simple mathematics, it's crazy to see the complexity of graph in later stages like for 4 or 5, not answers that we have to write down, but the graphs and formulas, the method to solve the problem is just vast and magnificent.

[@beautifulmath5361]

That is wonderful that you're exploring higher mathematics on your own! I'm delighted that you see the magnificence of mathematics already at age 13. You will discover so many beautiful things!

[@philipwatson2407]

This is probably the best visual representation of the topic that I have ever seen.

[@beautifulmath5361]

Thank you! I'm delighted you enjoyed it.

[@THEDeathWizard87]

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

[@beautifulmath5361]

Absolutely! Having the technology and the visualization tools is key to insights like the ones in this video. They didn't exist when I was in school, so my teachers can be forgiven for not teaching me these connections. Even when they do exist, teachers need training in how to use them, and more crucially, TIME to get to know them and create demonstrations using them. Even with all the tools at my disposal, I still often find myself unable to make demos like this for kids, simply because I'm too pressed for time with grading, lesson prepping and so on.

[@xl000]

people used to understand this without what you sugest.

[@ItsPouring]

​@@xl000 And people also used to use typewriters that didn't have memory, and instead of being able to port their phones with them so they could make calls while outside of the home, they paid to use a communal phone while in public if they had to make a call or had to wait until they got home, and most people memorized the phone numbers of those they called most...

[@masonseminario7435]

I am currently in calc 2 and still only understand the sin and cos lines even after watching this video

[@mtridenhour]

This demonstration of Trigonometry functions demonstrates the value of internet social media, KZbin. Let the viewers use internet search engines to find fighter jet skilled aerodynamic manuevering demonstrations, and see these Trig functions in action. Good job with this idea of teaching advanced mathematics!

[@wreckim]

As a teacher I believe that the earlier we just let kids watch and learn, the easier it's going to be for them. Imagine YOURSELF in a Chinese or German home at 6 months of age; you'd learn those languages completely effortless. This kind of visual is something we could all benefit from before we start getting confused and let our trepidation begin to overtake our God-given talents. Thank you for posting such a beautiful illustration.

[@beautifulmath5361]

Thank you! I'm glad this video resonated with you! I'm a CS teacher as well, and my students say all the time they wish they'd been shown things like this in math class.

[@greenrhinoguy]

This is beautiful. Math is so fun and interesting once you look past the stereotype of math class being boring and pointless. This video definitely helped me understand the unit circle definitions of the trig ratios!!

[@H3XED_OwO]

I completely agree :3

[@dereklenzen2330]

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

[@beautifulmath5361]

Wow, thank you!

[@estebson]

Bro just revealed all of trigonometry in a divine 8 minute video. Makes me wish I could like this video 100 times.

[@simpleman283]

💯,000

[@DarkifyDarkify]

I learn about triangles and angles from this video more clearly and better than I do from school

[@beautifulmath5361]

Thank you! I'm so glad it was useful for you!

[@spelunkerd]

I've been waiting for somebody to do this for decades. Bravo!

[@beautifulmath5361]

Thank you! I'm delighted it was helpful to you.

[@Firefin]

the fact that someone was smart enough to mentally visualize all of thid and understand the identities to such a deep level is insane to me

[@beautifulmath5361]

Decades ago I saw a diagram of the idea in a textbook. I thought it would be fun to make an animated version of it.

[@godoit7569]

@@beautifulmath5361 Could you tell that textbook's name,pls.

[@juanjuan5698]

I am doing math in uni. All the ppl who paved the way in mathematics are geniuses!

[@twopurplemoons]

Thank you so much for this beautiful trigonometry. I just wonder why is this channel is not popular. This video deserves views in millions.

[@beautifulmath5361]

Thank you for your kind words. Please pass it on to someone who you think might benefit from it. That's one way it could become popular! 🙂

[@lakshyaahir7232]

Apart from sin waves i struggled in other trig functions, now by watching the visualisation of it i scratched the surface of trigonometry. Thanks a lot.

[@beautifulmath5361]

Thank you for your kind words. I'm so glad this helpful for your understanding.

[@theoneobviousguy7328]

I wonder if this was how my father solved this stuff, my teacher taught me a very simple and easy way of remembering the sin and cos graphs and the other stuff, we started with sin, then cos, then simply, sec is 1 over cos and csc is 1 over sin, the tan graph has vertical assymptotes at 1 (90 degrees) AND THE TURNING CIRCLE THING ILLUSTRATED THAT AMAZINGLY! That wierd circular triangle with all the bits and pieces was so cool. W video

[@beautifulmath5361]

Thank you! I'm glad this video helped you connect those concepts!

[@AG-JustYourAverageGuy]

I never ever understood the quadrant rules of trigonometry and my teachers never cared enough to explain then this random yt video clears the doubt I have had for 2 years. Thank you from the bottom of my heart.

[@beautifulmath5361]

Thank you for your kind words! I'm delighted my video was so helpful for you!

[@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.

[@beautifulmath5361]

Thank you for saying so! I am delighted to hear this.

[@therealsuper5828]

same

[@jasonharvey3501]

Me too. I get it now.

[@fafablablabla]

You don't lie when your channel's name is Beautiful Math. Very beautiful indeed!

[@beautifulmath5361]

Thank you very much! 🙂

[@MatthewDickau]

Would love to see this for the hyperbolic trig functions - they have the same relationship to the unit hyperbola that the trig functions have to the unit circle, provided you measure distances with the Minkowski metric.

[@beautifulmath5361]

That is a good idea! Others have made the same suggestion.

[@ItsPouring]

Oh my goodness - GREAT idea!! 👍 In the meantime, you've inspired me to see what's already out there along those lines!

[@sicariochoarovin9643]

So beautiful! The music goes so perfectly with the images and evokes the wonder of the maths.

[@beautifulmath5361]

Yes, I picked the music for exactly that purpose. It's just a free track provided by KZbin, but it fits the theme. 🙂

[@carloskleiber2111]

Very Vangelis ...

[@beautifulmath5361]

@@carloskleiber2111 Yes, it reminded me of Vangelis, too! Sounds like the introductory theme in Blade Runner.

[@BerkeleyRadical]

100% described it as both blade runner-esque and vangelis-like to my BF a minute before reading these comments 😂

[@beautifulmath5361]

@@BerkeleyRadical Ha ha, good job calling it! And what better video than this one to curl up with your BF with. 🙂

[@paluz1177]

Damn this video is awesome. I never imagine what were sec csc & cot in the unit circle. The final refereence of sin sec tg in the same θ and the complementary functions (cos cosec cotg ) was incredible

[@beautifulmath5361]

Thank you very much for this!

[@Heikal.q]

Easily the most helpful math video I have seen on KZbin, thanks for making this.

[@beautifulmath5361]

Oh wonderful! Thank you for saying so.

[@georgeh1352]

Wow what a powerful video, wished we had this back in my school days (60yrs ago). Sec , cosec and cot always eluded me, I now have a solid understanding of how these angles work around the unit circle, no longer Trig Idents like (1+Tan sq theta = Sec sq theta) mystify me. Many many thanks, you're never too old to learn!

[@beautifulmath5361]

Fantastic. I'm delighted this video helped consolidate your understanding!

[@Gunslinger-us1ek]

I returned to this after 4 months as now only I have realized how useful this is. Thank you so much!!!

[@beautifulmath5361]

You are very welcome! I'm delighted that this video was helpful for you.

[@gove4103]

I'm old, and an engineer, and this is the first time I've ever actually fully understood the secondary trig functions.

[@beautifulmath5361]

I'm delighted this was helpful for you, even at this late stage in your career! I'm well into the 2nd half of my career too, and there are still things I am learning that I wish I'd seen when I was younger.

[@ratguy278]

this is so awesome. the music especially turns it into an almost meditative experience. it feels like im on another plane of existence

[@beautifulmath5361]

Thank you! I'm delighted this video struck an emotional and meditative chord with you in addition to a mathematical one.

[@kutlass]

This is the first time I've ever gotten goosebumps watching a math video

[@beautifulmath5361]

Oh wow, that is the best compliment ever. I'm delighted you enjoyed the math in this video

[@jdoesmath2065]

Such elegant relationships. And presented so beautifully. Thank you.

[@baselinesweb]

This is really well done. I've never seen the identities mapped out like this.

[@beautifulmath5361]

Thank you!

[@TheFireBrozTFB]

O man, similar triangles, so elegant yet unknown when learning trig.

[@beautifulmath5361]

Astounding that similar triangles aren't taught more thoroughly during trig! Similar triangles is what makes sin, cos and tan even possible.

[@TheFireBrozTFB]

@@beautifulmath5361 thats truee! Very sad that many people have missed and will still miss out on these ideas

[@RJNoe]

This video is truly deserving of the name of this channel. Beautiful math, indeed! Very nicely done.

[@beautifulmath5361]

Thank you! So glad you enjoyed it.

[@simpleman283]

I could not agree more.

[@traitorslayer1798]

Video was awsome i now have a new understanding for trig as i always say understanding whats behind the math is the key to success and comprehension

[@beautifulmath5361]

I'm so glad this was helpful for you!

[@exurbian2420]

I have never understood the spacial relationship of the inverse functions before. this is a beautiful video that every student should be shown

[@beautifulmath5361]

Thank you! I'm so glad this was helpful for understanding the reciprocal functions!

[@kounova]

I've finally got it, after so much time of not understanding trig functions, this feels like forbidden knowledge

[@beautifulmath5361]

That is fantastic to hear! A wonder that this "forbidden" knowledge is not standard curriculum.

[@DifferMeme]

Beautiful

[@beautifulmath5361]

Thank you!

[@hyper0699]

The perfect video to watch before my math exam.

[@beautifulmath5361]

Fantastic timing. Let me know how your exam goes!

[@XavieRFCB2004]

@@beautifulmath5361 i got 99.68 in advanced trigonometric function math the only mistake i made was making the cot=sin\cos and it was wrong so yeah thank god for everything

[@sebbes333]

5:25 Alternatively: The "CO-functions" lives along the vertical line. The other functions lives along the horizontal line.

[@luisdmarinborgos9497]

This video is gold. It should be shown in every school to people first learning trig. You got yourself a new subscriber 💯

[@simpleman283]

Now almost anyone can see it, if we share.

[@beautifulmath5361]

Fantastic! I'll be sure to make more!

[@jookie2210]

I learned that the tan is tangent to the circle just like the cot but in the other direction and the sec is similar to the csc but it is the distance from where the tan hits the x-axis to the origin.

[@Deathkyun]

When I was still studying, I was really good at geometry and piss-poor at trigonometry. In geometry, everything just kind of presented itself in a way that made sense and answers weren't really that hard to figure out. Trig I found really difficult. Getting a handle out of all the rote things to memorize, and then to try and apply them to problems? It was easy for myself to get lost in trying to find answers. So having this just SHOWN to me now, I'm kind of speechless. These animations made trig just as intuitive for me to see as geometry did back then. Hard to feel that it wasn't kind of a waste not knowing how easy it could have been for me all those years back, but I certainly hope this finds another kid out there who was in my position, and makes good use of this. Well done, man.

[@simpleman283]

Well said.

[@beautifulmath5361]

Thank you! I'm so glad this was enlightening for you.

[@celedoniojimenez-ww1tb]

Ohne Schäme bekenne ich hiermit, daß höchst beeindrückt bin. Herrliche Vorstellung der Nummernwissenschaften.

[@eduardo33]

Amazing, it`s like a explosion in the brain, BOOOOMMMM!! You explain these functions at 8 min. A classroom will take several Hours. You are the best!!!!!

[@beautifulmath5361]

I'm glad it was helpful to you!

[@unknownhuman9416]

I've always been an average student in mathematics, but surely I was most scared of trigonometry. But this is the first time, I feel this is special. This is a beautiful explanation.

[@beautifulmath5361]

I'm so glad this helped you get over your fear! :-)

[@marcoarodriguez]

This video helped me visualize the trig identities. I am starting Calc 3 next week and didn’t know until now how they were related using the Pythagorean theorem. I always wondered why they were squared or you add/subtract one. I also enjoyed watching the trig functions graphically. Very interesting video.

[@beautifulmath5361]

I'm so glad this was helpful for you!

[@mavairick]

Never forget that once, we didn't knew about these and some guy just drew some circles and lines and decided to find a formula to calculate angles and length, and they just did it out of their mind. We have the incredible luck to be told these solutions, formulas, theorems, we may not use them in every day of our life, but it's worth knowing that the day we need it, we'll already have it.

[@beautifulmath5361]

Agreed! As Isaac Newton said about himself, "If I see farther than others, it is because I've stood on the shoulders of giants."

[@RealGeorg3]

I've been using trigonometry for years and years. This is an excellent visualization. I'm certain it is helping a lot of people. Thank you.

[@beautifulmath5361]

You're very welcome! I had fun making it.

[@LJ_Brostep_Productions]

I am almost brought to tears over how much I have learned in this 8 minute video, it is really beautiful to see what you have made here. Thank you for this.

[@beautifulmath5361]

I am delighted to hear this! I am in tears with you.

[@silentcartographer7685]

Best visualization of this I've seen thusfar.

[@beautifulmath5361]

Aww,thanks!

[@moonwatcher2001]

Really well done, beautiful, interesting, with amenity, intuitiva, useful... And relaxing music. Thanks mate!!!

[@beautifulmath5361]

Thank you! I'm delighted you enjoyed it.

[@Wallyisking]

Educational material like this can help some students learn in minutes what may have taken others days. It's amazing to see how we continue to improve on our own learning methodologies as a society!

[@beautifulmath5361]

I fully agree!

[@The_Acquainted_Era]

Stunning. Maths meet art and I'm constantly laughing at myself for not watching this video till today. Great work

[@beautifulmath5361]

Thank you, I'm delighted you enjoyed it!

[@nug700]

100% showing this to my Calc and Pre-Calc college math teachers on Monday.

[@beautifulmath5361]

Fantastic! Pls let me know what they think of it.

[@calor5010]

This is really cool. I think everyone learning trig should be shown this.

[@simpleman283]

💯%

[@DazHuang72]

This is single video should be viewed by teachers and students together and have an active discussion together when learning about trigonometry.

[@beautifulmath5361]

Thank you! I made it exactly for this purpose. 🙂

[@Googus1]

The music induces a sense of calm

[@beautifulmath5361]

It's just a free track provided by KZbin, but I like it for the same reason. 🙂

[@seanhayes2998]

This video taught me more in 15m than 5 years of school maths. Shame it took 50 years to find it!

[@beautifulmath5361]

😅 What a tragedy it took so long, but I am delighted you found this video useful!

[@awesomecronk7183]

Matt Parker has blown my mind before but this set of animations just blew it all out of the water...

[@beautifulmath5361]

Wow, thank you! I like Matt Parker, too, so this is a fabulous compliment.

[@solcarzemog5232]

Hypnotizing beauty of geometry and math, I just realized how little I knew about the CO-functions. I'll watch it again and again.

[@beautifulmath5361]

Yes! Teachers seldom talk about what the CO- in cosine, cotangent & cosecant actually means.

[@ZMan778]

Was am I so amazed at this? I’m literally in Pre-Calculus, we’ve been doing trigonometry for like 2 months now so I really shouldn’t be surprised at this… But just seeing it laid out like this with these animations is just so cool to me.

[@harrygenderson6847]

Only 2 months? I first did trig 7 or 8 years ago. I knew the sin, cos and tan representations on the unit circle, but not the rest...

[@ZMan778]

@@harrygenderson6847 are you in college or high school?

[@harrygenderson6847]

@@ZMan778 college

[@ZMan778]

@@harrygenderson6847 I’m still a junior in high school

[@randomz5890]

This is such a beautiful video. Would single handedly make so many more people interested in trig.

[@beautifulmath5361]

Thank you so much! This means a lot.

[@subashshanmugam5411]

This teaching methodology was absent for decades in TN schools. This graphical illustration with enthralling music makes Maths learning a passion.

[@beautifulmath5361]

I'm so glad you enjoyed this! I agree about teaching methodology. I wish I'd had something like this growing up, too.

[@inven2win]

I have envisioned how to plot sin and cos as a projection of a point on a revolving circle (vary the angle), but thanks to this animation I finally understand why they call the third function "tangent." There are so many wonderful graphic illustrations for math that you can find on the Internet! Great one... and so simple.

[@beautifulmath5361]

I'm delighted you found that part insightful!

[@aalexazander]

This is so satisfying to watch

[@agentember2143]

From my junior year precalc class, the one thing i took away from trig was i hated trig because of the trig identities. Now that I'm in a college precalc class, this is genuinely extremely helpful for being able to memorize how they work woth each other and derive identites from there.

[@beautifulmath5361]

I'm so glad this video helped you appreciate trig more!

[@zhubwat]

I'm a maths tutor, and it's very refreshing to see so many people just enjoying trigonometry, for once. :P All jokes aside, it's a very impressive animation. I think to some degree, many people know mathematics, but less people "get" it. Many students know the tools, but not what they mean - and it's not just trig. Algebra, on account of being algebra, is plagued by that. Really happy to see you so engaged in the comments, too. Excellent work!

[@beautifulmath5361]

Thank you! That means a lot! You are spot on about algebra having plagued by the same communication problems as trigonometry.

[@cjayy1]

This made me cry.

[@beautifulmath5361]

Hopefully in a good way!

[@zo0bit]

I wish I had this a decade ago. Trig is the one part of highschool math I could not wrap my brain around and this would have helped so much.

[@beautifulmath5361]

I agree! I wish I'd had this myself when I was in high school 30+ years ago. I once saw a diagram in a text book that showed the main configuration shown in the thumbnail. Once I learned how to code, I got the idea to create an animated version of it.

[@samwisegamgee4659]

I wanted to cry this was so beautiful.

[@beautifulmath5361]

Aww, thank you! I too amazed at how breathtakingly beautiful mathematics and geometry can be.

[@H3XED_OwO]

Thank you so much This is why i love math; All these different parts acting together in a relationship, it gives a sense of harmony. (especially with this music)

[@beautifulmath5361]

Thank you, I'm delighted you enjoyed it and that you love math like me. The music is just a free track provided by KZbin, but I like it because it fits the ethereal nature of this kind of math.

[@dbqp17]

Thank you, your video made me cry. For the beauty of its content. The universal language, gracias, grazie, danke, obregado, merci, spasibo, tashakhur, shukhran, shukhria, arigato, shishe, etcetera, etcetera.

[@beautifulmath5361]

Awww, I am delighted to hear this! Thank you for your kind words.

[@alfredolosacco2665]

This is most beautiful math animation ever, thank you so much for your dedication.

[@beautifulmath5361]

Thank you for your kind words! So glad that you enjoyed it!

[@WilliamWilliamWilliamWill]

Literal perfection. Thank you for this masterpiece. It feels like bringing my imagination into life💙

[@beautifulmath5361]

Aww, thank you for your kind words! I'm delighted this was eye opening for you!

[@dominicestebanrice7460]

Stunning to see so much of the universe represented in one short video! Amazing how a unit circle, similar triangles, Pythagoras & trig. are at the heart of waves, and therefore electromagnetic radiation, and therefore energy, and therefore EVERYTHING. Vibrations, wave functions......on and on, all emanating from the fundamental relationships presented in this masterpiece. Thanks for sharing this, it is one of those creations that makes dealing with al other shit that we have to wade through online all worthwhile. And the music was totally appropriate IMHO.

[@beautifulmath5361]

Thank you for your kind words! I totally agree, trig lies at the heart of all understanding in physics. Glad the music resonated with you as well! I found it fit the ethereal nature of the mathematics it's accompanying.

[@heatstorm9312]

Never fully understood why these functions are what they are but this video makes it so much easier to visualize and understand

[@beautifulmath5361]

I'm so glad it was helpful for you!

[@jakob2478]

until this point, i did not know that math can be so beautiful to watch😢❤

[@beautifulmath5361]

Thank you! I'm so glad you found this moving!

[@tunneloflight]

The second way of looking at it is so very much more intuitive. Thank you.

[@beautifulmath5361]

Good to hear this feedback! Thank you.

[@mx.fuzzypants1911]

This is beautiful. I’ve looked at this for five hours now.

[@beautifulmath5361]

Thank you! I'm delighted that you like it so much!

[@aikiiru]

Goosebumps... Thank you for your work!

[@beautifulmath5361]

You are so welcome! I'm glad this video was inspiring for you!

[@SirDooms]

This needs to be shown in every classroom. Wow

[@beautifulmath5361]

Thank you for this! I wish I'd seen something like this myself when I was in high school 30 years ago.

[@Davidparty3]

What a beautiful way to visualize it, amazing

[@beautifulmath5361]

Thank you!