All 6 Trig Functions on the Unit Circle 

That is fantastic to hear! Thank you!

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

​​@@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!

@@andrewsemenenko8826 Excellent point! 🙂

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

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!

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

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

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

​@@cmyk8964 same for me, thank god

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

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

Music needs work, though

Can recommended the Mandelbrot set animated

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

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

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

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.

@@beautifulmath5361 that's really cool

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

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.

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.

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

Thank you!

Is that what the CO means?!

​@@Quroe_ 5:50

​@@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.

math is beautiful

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

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

Great synths I love the correlation math has with music

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

For sure, there could be nothing without math.

Beautifully said! I agree wholeheartedly!

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.

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...

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

@@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...

@@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! 💜🌎📐🍀

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

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.

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

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

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.

people used to understand this without what you sugest.

​@@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...

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

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

same

Me too. I get it now.

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!

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

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

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

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

I completely agree :3

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.

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

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

💯,000

Wow, thank you!

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.

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

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

Thank you!

Thank you! I'm delighted you enjoyed it.

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! 🙂

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

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

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

Very Vangelis ...

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

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

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

The best mathematics has a hypnotic effect! :-)

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

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

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

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

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...

@@harrygenderson6847 are you in college or high school?

@@ZMan778 college

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

I'm glad it was helpful to you!

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

Now almost anyone can see it, if we share.

Fantastic! I'll be sure to make more!

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

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

Thank you for sharing this story! I'm glad that you sought to use math for a very practical purpose and that you found this video helpful!

Oh wonderful! Thank you for saying so.

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

Thank you! I'm delighted you enjoyed it.

Well said.

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

I think they go hand in hand. A live teacher is needed to introduce the core ideas and motivate the reasons why we study math, as well as teach the mindset of a mathematician. Videos like this one and others add to that by building a visual and spatial intuition that is hard to replicate on a static writing surface.

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

Thank you!

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

I'm so glad this was helpful for you!

Yes! Fantastic that you anticipated this!

@@beautifulmath5361 This is a really fantastic (and beautiful) video. I did well in mathematics through this level and far beyond, but in school I was NEVER shown graphically that tan, cot, sec, and csc have actual geometric meaning. Thank you!

Thank you! So glad you enjoyed it.

I could not agree more.

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

Thank you for your kind words! I'm so glad this was useful for you!

💯%

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

I'm delighted this was helpful to you.

Thank you very much! 🙂

I'm delighted you found that part insightful!

I fully agree!

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

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

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

Thank you, I'm delighted you enjoyed it!

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.

And better than what a person can do on a plain white board

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

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

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

Thanks for sharing that story!

Thank you for this! This is precisely why I made this video. I would have benefited from seeing it taught like this when I was in school 30 years ago.

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

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

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.

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

I'm glad you like it! It's just a free track I picked from RU-vid'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.

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

Thank you! So glad you enjoyed it.

Fantastic to hear this. I've been teaching high school computer science and mathematics for 16 years, and I am still learning new things about these topics, even ones first taught at the high school level.

You're very welcome! So glad you enjoyed it.

I'm so glad this was useful for you! So sorry to hear that your earlier courses didn't do a good job showing you these important connections!

I am delighted that my video was helpful to you! Many commenters have asked for a video on the hyperbolic functions, so I think I will! Stay tuned.

I'm glad that visualization was helpful for you!

Thank you so much! This means a lot.

Wonderful! So glad to hear this brought things together for you!