Beautiful Trigonometry - Numberphile
Vložit
- čas přidán 15. 06. 2020
- Featuring Ben Sparks.
Check out Brilliant (get 20% off their premium service): brilliant.org/numberphile (sponsor)
More links & stuff in full description below ↓↓↓
Catch a more in-depth interview with Ben on our Numberphile Podcast: • The Happy Twin (with B...
Ben's website: www.bensparks.co.uk
More Numberphile videos with Ben Sparks: bit.ly/Sparks_Playlist
Explore this topic further with this Geogebra file from Ben: www.geogebra.org/m/S2W46Thv
Catch Ben on the latest Numberphile Podcast: • The Happy Twin (with B...
More podcast episodes: bit.ly/numberphile_podcast
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoundation.org/outr...
And support from Math For America - www.mathforamerica.org/
NUMBERPHILE
Website: www.numberphile.com/
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberphile_Sub
Videos by Brady Haran
Animated by Pete McPartlan
Patreon: / numberphile
Numberphile T-Shirts and Merch: teespring.com/stores/numberphile
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanblog.com/
Sign up for (occasional) emails: eepurl.com/YdjL9 - Věda a technologie
Catch a more in-depth interview with Ben on our Numberphile Podcast: czcams.com/video/-tGni9ObJWk/video.html
Numberphile Late to the party I know, but the final set of animations is how AC motors work, as well as field oriented control for brushless DC motors. With three intersecting lines it shows the relationship of three phase mains power. Most motors only use 3 phases, but multiphase motors do exist. Would have been cool to explore in further detail - maybe a future video?
captions in spanish please!
This video needs to be shown in every high school trig class
Yes please. :)
I can still remember memorizing sin and cos values when I was in high school almost 30 years ago. It would have been so much easier to have been taught circle functions and tau. Why memorize when it's so easy to derive. So much time wasted 😢. At least younger learners will have an easier time.
That said, I thought I would show these things to my kids and they will see the awesomeness, but I've had bad experiences so far with responses of "that's not what the teacher expects", and it makes me very sad 😞.
@ "that's not what the teacher expects" that is the dark side about teaching
@ it's a sad time we live in where surprising the "teacher" is seen as a bad thing.
I never took Lockhart seriously until I tried to help my incredibly creative little brother with his homework, and saw all his mathematical imagination completely stamped out of him by modern maths education.
i wish my teacher showed me this. before this video i think of trigonometry as a formula that just works. Now i understand it a bit more.
"and I don't want to give too much away"
Brady: So I'll just spoil it in the thumbnail and intro then.
Ben doesn't have access to the graphs which show CZcams viewer attention span - Brady does!
@@numberphile
Oh, I'm sad to hear that. I thought Numberphile's views were 90% subscriber users who watches everything. : (
@@numberphile the Algorithm...
@@HasekuraIsuna were drawing in new fans, get them all addicted to math, then they become full length viewers
to 10 anime betrayals
"I think [trigonometry] is the worst-named topic in mathematics"
Imaginary numbers would like a word...
Imaginary numbers don't have to be a topic. They just have to be numbers. It's up to the speaker to set it as a topic.
Aren't all numbers imaginary?!
@@danaclass Aren't all imaginations numerical? :O xD
@@danaclass I mean vision is technically, an elaborate elusion. So sure why not.
My thoughts EXACTLY
I learned more trigonometry in this 12-minute video than I have in 40 years...
Those animations are mesmerizing
bruh when did I leave this comment
@Multorum Unum fair
You could make one yourself, just go on desmos and the description for a point revolving around the surface of a circle is (sin a, cos a) With that info you can add them up in specific ways and have lots of fun. I hope you try it our
@@phatkin 3 days ago
I agree
Something about the patient, thoughtful explanation combined with insightful interview questions combined with excellent, simple animations make this the best explanation of trig I’ve ever seen.
Wow, thanks.
agreed
I couldn't agree more! 30 years after I leave school and suddenly it all makes sense!
So true!
It's still beyond the understanding of a certain segment of our society though, and that's sad.
This just made some of the bits of trig I just didn't get fall into place. Ben Sparks needs to rewrite how schools teach trig! That's was brilliantly explained!
Brought to you, by... Brilliant.
arcanics1971 uhhh my last name is also sparks lol
arcanics1971 and im also a yter
That's Class 11 physics its taught in schools
3:35
Complex/Imaginery numbers
am I a joke to you?
This annoys me quite a lot. I'm 36. I got great grades for Chemistry, Physics and Biology at GCSE but did abysmally in my maths exam and the best grade I ever achieved is a D. The exams at the time were full of trigonometry and I have NEVER been able to get my head around that one thing all my life.
Seeing it applied to circles, rather than triangles has literally allowed me to understand the subject in the length of this video.
I wish you were my GCSE maths teacher 😂
When it comes to school level mathematics, your success is mostly based on repetition of work, not understanding. But yes, this is a brilliant visualization of the trigonometric functions.
@@bregottmannen2706 "SohCahToa"- not all teachers teach
some just need you to be able to pass a standardized test at semester's end
I am a maths teacher and hate that the curriculum focuses on triangles simply because that's what they are applied to in the syllabus.
Circles are brought in when learning about the graphs but it's constantly compared to the triangles (because they've already studied them).
It would be so much better (and elegant) the other way round but everyone learns about the triangles at GCSE and only students sitting the higher papers learn about the graphs.
I also wish my maths teacher had described things like this!! I failed Maths and stopped it after 5th form in NZ.
But this is beautiful!!
@@scottriley5141 Porque no los dos? The triangles and the circles in this case are inherently connected when you're using the cartesian plane or space (you can always draw the triangles for any point in a circle, you can always draw the full circle from a right triangle without further information, which is why they're equivalent). The problem is that the syllabus and education in general don't exist to teach people to understand what they're doing, just to be able to memorize how the tool operates. But when people understand the relationships, they understand where things come from and the line of reasoning that leads to it, you see the triangles on the circles, you see the circles around the triangles, you can see all the other forms you can derive from the same information because you understand. That is why no curriculum anywhere puts much attention in showing things like this, the beauty of pure mathematics because "it is not useful [to make a baseline employee, the curricula current purpose, sadly]".
"It was a huge relief to me too that the word tangent wasn't a coincidence with the other definition of tangent, which touches the circle" 😇
4:50 In fact the name cosine comes from the fact that the **co**sine is the sine of the **co**mplement angle.
Two angles are complementary when their sum is 90 degrees. So 30 degrees and 60 degrees are complementary angles.
The cosine of 30 degrees is the sine of its complement angle. That is, the cosine of 30 degrees is the sine of 60 degrees.
interesting
Yes. Hence the origin of the cofunction identities sin(π/2 - x) = cos(x), sec(π/2 - x) = csc(x), and tan(π/2 - x) = cot(x). π/2 = 90°. This also motivates the conjugate identities sin(x)·sec(x) = tan(x) and cos(x)·csc(x) = cot(x).
Ah yes big brain you have
_82_
One of the best Numberphile videos I've ever seen. Coming from a middle/high school math teacher. Thank you! Keep up the great work!
My overall happiness with maths in school would have increased dramatically if this channel existed (and was in German) back then.
Deutschland oder österreich?
That 3D graph with the spiral changed my view of Trigonometry - truly beautiful. Love the work as always :)
Thanks - appreciate it.
Ikr, my jaw literally dropped to the floor when the changed the perspective
I can finally see the dual particle/wave nature of light in this!
Helix
Numberphile
If you replace the y-axis with the imaginary you’ll have Euler’s Identity. I know you know that. It would have been a great comment to make during the video.
I also believe mentioning the deep connection to light propagation by allowing the x-axis to be a voltage field and the y-axis a magnetic field yields the way light works propagates would have been relevant.
Oh, and when you showed the three axis’s with three dots rotating you had three phase power which is how the power grids around the world distribute electricity. Again, you showed the answer without mentioning the topic.
Totally fun. Maybe I need to learn Visual Basic better to be able to reproduce your animation here.
How many of the trig identities make more sense on this graph? I’m thinking they may all be here, but several of them seem to just fall out.
wow! This is a great example of the beauty in maths. Great teaching with a great explanation of the animation. Loved it.
Thank you for watching it.
@@numberphile I agree :D this is beautiful and well made, I love this channel!
4:57 I *wish* I had seen this when I was still at school. Why was trigonometry never explained to me like this? It suddenly makes total sense now!
Because at school you don't need to know it and it'll probably just confuse kids. The sad thing about school is you only learn the basic application of things, not the actual mechanics and processes behind them
@@jadenmolloy4830 yeah, it's *general* education, higher education is for the real learning and school's just the basics
*the definition i like most is: oscillation is the projection of uniform circular motion on the diameter of circle*
the simplicity is amazing!
@@hoola_amigos I agree, just realized what trigonometry is really about
AWAZ2
I dont if what you say is true- for all I know is that you're saying big smart-sounding words
A lot of math concepts come to ife and thus make more since in physics. I.e. calculus made much more sense to me when I saw how it applied to kinematics
@@mr.klunee4103 yes✔
Ben: 'Nothing is moving in a circle'
Me: 'Wait, so how does the earth spin?
Ben: 'Each one of these dots is moving in a straight line.'
Me: 'Oh, right. Context.'
😂♻️
Careful there... We don't need to be spending the next year explaining to flat earthers how this is in fact NOT evidence that the world isn't a globe 😂
The Earth is flat so there is no reason for anything other than straight lines!
The is only works if the radius of the moving "circle" is equal to the radius of its orbit so no
I see a new understanding of General Relativity in the making. There is nothing that doesn't move in a straight line, relatively.
I love how 10 minutes of video can give me a real understanding of what sin, cos, and tan *are* when none of my high school teaches ever thought to try. No wonder I hated trig...
Yeah, because your teacher never had to spend any time/focus on behaviour management, admin, catering to different levels of understanding of students, time reatraints, did they? And you always paid attention 100% of the time, didn't you?
Funny how commenters almost always take the opportunity to blame teachers, thus abdicating any responsibility themselves.
@@Jinsun202 What is blud waffling about?
A straight line is just a circle with an infinite radius
Random Jin a circle, in turn, is just an ellipse with 0 eccentricity
@@F1fan4eva An ellipse, in turn, is just a conic section with eccentricity between 0 and 1
@@harry_page a conic section, in turn, is just a curve obtained as the intersection of the surface of a cone with a plane
What if a straight line is just a conic section with INFINITE ECCENTRICITY
@@Killerthealmighty It is true. We just loop back to where we started. both eccentricity 0 and infinity gives us a line
6:37 "What's a tangent?"
Perfect answer would've been, "What you're doing right now."
OMG!!! This video just blew my mind. The trig functions make sense in relation to a circle. Dude you just made me unlearn what have learned about these functions for real.
this cleared up so much missing understanding
"And nothing here is moving in a circle" ... I mean except for the invisible point that you're using to trace out the coordinates of the dots :P
Some left over 'Pieces of Pi' are available if you'd like to support Numberphile (these are from the collection that was sent to Patreon supporters earlier this year): posh-as-cushions.myshopify.com/products/a-piece-of-pi
Piece #1 is on ebay: www.ebay.co.uk/itm/203022020704
The animations were awesome!😍👌
I understand 'trigonometry' better than ever before!
What is the relation between (ΠXi^xi(i,1,n)) to (ΠXi(i,1,n))^y ,where, n is Z+ ,and y =f(x).?
What’s the missing sin()-like function called?? It’s clear when you overlay sin() and cos() that they are out of phase, but there’s space left over for one more phase to perfectly fit in between?!?!
May I ask what program Ben is using to create these animations
That 3D animation really is beautiful
0:13 to see straight lines concentrate on one ball moving up and down and then the other and look in the centre
To see the circle concentrate on the invisible line between the circles
I actually never learnt trig in school. The first time I ever used trig functions was when I wanted to draw a circle on a screen.
X axis is the cosine of t, Y axis is the sine if t. Now just take equidistant values of t between 0 and tau and you're golden. More values for higher resolution.
This should be the first thing shown in school when trigonometry is mentioned.
No, it shouldn't.
@@pendragon7600 yes, it should
Maybe not this _exact_ video, since it definitely assumes some prior understanding of trigonometry, but the visualizations definitely.
@@SSM24_ true yes. but this animation is easy to grasp for visual learners with minimal equations to have to learn.
It's a great way to pretend you understand trig without actually learning anything. This is much better appreciated AFTER you actually learn the concept and use it properly.
10:08 this tells me a lot of why my math teachers tend to make really hard tests...
They make hard tests so you have a reason to study. I'd bet you would open the book once if you didn't have to study for a test
You are the same type of person to get a bad grade in optimization problems in calculus then say "we did not learn this! You gotta spoon feed me!". And then you realize its useful, applied math.
This taught me more than 3 years of highschool. I want more. I think you healed my soul with those mathematical animations. I actually hate how much I liked that. This should be a series. Just mathematical animations and the proofs hidden within.
I've watched Numberphile for years, and although I've always enjoyed and learned something from the videos, it's never impacted me quite like this. The simple look behind what sine and cosine really are was a wonderful experience.
Thank you.
The fact that ramanujan could see all possible combinations of these functions in his head still boggles me
I'm more than a little confident a lot of the Ramanujan stories are legends. This idea that he could just "see" everything is kinda silly. That he just knew strings of trivia for every single number, etc.
@@ZachGatesHere there are some really interesting video's on youtube about savant's who 'see' numbers and can do crazy sums in their head, maybe that is related, you should check it out if you haven't already.
@@ZachGatesHere when it comes to trigonometry he was a beast. Before he read Carr's book he had mastered Looney's trigonometry. He didn't steady Euclid's elements so I can't be sure of his mastery of geometry and in general but there is no doubt about his understanding of circular functions. Just look at his approximation of pi.
@@ZachGatesHere not "see", but "imagine". although Ram sadly is not that great at explaining the theories behind it. He's really just your more-than-average smart kid in class who understands stuff and can solve puzzles, but has a hard time in explaining the process behind. Don't get me wrong, Ramanujan is still a big part of why modern life in general exists, that's his legacy.
These animations are so spectacularly beautiful, all the jigsaw pieces just flitted right together 😭! Pretty exceptional stuff, wish school taught us this way.
Never stop these videos coming, your work is amazing!!
““A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” A mathematician apology Hardy
thats a beautiful quote
Of all the Numberphile's brilliant mathematician talks, I love Ben Sparks the most. Every single episode is fascinating.
In case anybody wants to read more about this, the yellow circle and the unit circle make what mathematicians call a “Tusi couple”, and in fact, it does have some applications, especially in astronomy.
Just what I signed in to say. I first ran across it in a book on Arabic astronomy.
"Nothing is moving in a circle"
*Sad carousel noises*
nice
But it is moving in a circular reference frame cycloid, in the corner case where it became a straight line (rolling diameter equals enclosing radius). Another corner case is where it becomes immobile, because the rolling and enclosing circles are equal. All the other cases are actually useful in designing e.g. cycloidal or planetary gears!
@@0LoneTech Haha circles go woosh
@@diegonals mega nice
But if you put a couple carousels on a larger carousel, then timed the speeds right, could you get all the seats to just be moving in straight lines?
After twelve years of school, four years of university and eight as an engineer, I’ve not thought of the ‘trig’ functions like that. This is why I subscribed to Numberphile!
2:22 Trammel of Archimedes is mentioned (somewhat in passing). This is a topic worthy of visual elaboration.
The trace at the midpoint is shown, and it, indeed, is a circle. Both semi-axes are equal (thus = radius).
The family of ellipses created as the trace point is varied, including onto the trammel's extension, I find of interest.
Considering the name "trigonometry", other languages use the term "goniometry" more commonly, moving from triangles to just angles.
Trigonometry originated at the Indian subcontinent receiving important influences from the Middle East and it picked it up various implants in its name meanwhile.
@@azureabyss538 it may very well have originated on the Indian subcontinent, but the _name_ "Trigonometry" sure didn't. That word is as Greek as you can get.
@@silkwesir1444 I didn't mention that the name originated there, did I?
What a great way to explain it. If that had been presented to me in primary school I would have been light years ahead.
“Oh yeah it’s useful, but I would have done it anyway” is the best quote to describe a mathematician
Guys, your channel is gold. I'm repeating myself over and over again.
1 year of geometry class in 10 minutes. Awesome
3:35
Complex/Imaginery numbers
*am I a joke to you?*
More like
"Understand my reality"
I still debate if the name “lateral units” would have been any better. Read somewhere that was a debated name. That said, I can’t think of another possible name for complex analysis, or anything dealing with a name for the algebraic completion of the reals!
I mean, they aren't wrong, complex numbers are pretty complex.
AlisterCountel The name lateral unit would apply to the imaginary unit, not the set of complex numbers in its entirety.
It’s amazing the amount of work and love that goes into these videos, these animations are beautiful!
I loved hearing Ben on the numberphile podcast, he’s encouraged me to try for a PGCE after undergrad
this guy is an amazing teacher please have more videos with him!!
My calculus teacher in highschool always said "everything you need to know about the trig functions can be found in their graphs." Guess he was right.
I would love to see a follow up video on Trigonometry. It is one of my favourite fields of Mathematics at school. Maybe some beautiful proofs for the identities among other things would be wonderful. Mr. Ben Sparks is really talented and passionate and I love every Numberphile video which features him!
How about a video on hyperbolic functions and their geometry?
one of my fave teachers!
strangly, but I have learned that all of these properties of sin/cos at my school in Russia and we had exactly that circle and even more advanced ones to help us in exams
Why are the Ben Sparks videos the best on this channel?!
The different interpretations of the basic version are so beautiful.
Would be really cool to see these animations also with the hyperbolic functions!
I know I'm late to this party, but yeah - that's exactly what I was hoping for after he covered secant, cosecant, etc. I don't think you can use a point moving around on a circle to reference the hyperbolic functions, the way everything was shown in this video though. The shape that the reference point would move along would have to be a hyperbola instead of a circle.
That IS beautiful. Simple yet brilliant video.
I knew all of these things, and have seen a lot of the visualisations before, but seeing it all in one place and having it all synthesised together is awesome.
Having the sound phase with the opening few animations was an excellent touch
"it's got no use" - creates ball bearing from first principles
And other useful things like Wavepropagation, Gear(set)s... Ok, everything has additional details. And I am sure these principle ist used even in way more cases.
Maybe it's got no use for Mathematicians ;-) :-D :-D
10:001"That's the bit that's got no use". How about for converting rotary to linear motion in Mechanics or visa versa ?
you are spot on...without this mechanic the piston driven internal combustion engine or steam engine would not be possible
I don't think it would have been very fitting for him to say "and this is the bit that's useful for converting rotary to linear motion in Mechanics or visa versa"
It's just his way of saying that it's the end of the trig lesson, you can't learn much from this any more, it just looks nice
This takes me back to my early 90s Atari ST demo scene days. I had a lot of fun with modulating multiple sin/cos waves with different amplitudes to make some quite pretty animations. It's great that the tools exist now to explain this stuff so clearly.
Beautiful, both the mathematics and Ben's passion teaching it.
If Ptolemy made trigonometry it would be like this.
i think this is the most educational video i've ever seen. i never learned trigonometry in high school (not sure if it was skipped or not even in the syllabus) and i still don't know how i passed my calculus exams for computer science.
9:54 I love how Brady's Aussie-ness is coming through when he gets excited
Every bit in this episode I knew already, but it was a wonderfull version to visualize it to my kids, when talking about it. Thanks for that.
Keanu voice: "I know Trigonometry"
The instant I saw that first animation I knew it was sin and cos!
I seriously love your guys for making these videos. I literally watched all videos on this channel multiple times. My favourite ones are those about really large numbers, Surreal numbers and my most favourite topic of all INFINITY ❤❤❤. I cant thank you guys enough ❤
Definitely recommending this video for my calculus based physics students. This is perfect for a review of vectors, this lends itself well to simple harmonic motion, great for polar coordinates, and eventually complex numbers. So dope!
You: gets 3.14 subs
Me: Oh yeah, it's all coming together
This is very relevant to my current research into the (iterating) fractal, thank you.
Another comment to say the same thing as the rest: this video was so incredibly helpful and inspiring. Visualizing sin and cosine waves on 3 planes cleared up the entire concept of trigonometry very nicely. I wish this type of video could be shown more often in public schools. The in-depth (and fun) explanations that come from this channel could do so much to keep students interested and confident in mathematics. Thank you for the content, I will definitely be watching this video again.
I saw a connecting rod on a piston driving a flywheel moving with the motion of one of these dots. A very elegant piece of engineering and now I know how the motion is constructed. Thank you. I wish my teaches at school had described the trig functions as you have.
0:19 Cool overtone background. Almost sounds like someone singing through the harmonic series or something. Nice maths too tho hehe. :)
"mitochondria is powerhouse of the cell."
This makes so much more sense now!! Intuitive clarity, if that's a thing!! Excellent!
I love this. It explains the underlying concept rather than just "How it works". I feel like understanding why something is the way it is will better help accept it and understand it.
Coincidentally my coaching also started trigonometry woooooooooooo!
*I want the making of this animation*
I believe that this was the most useful videos I've seen.
I literally got the shivers.
I mean each Numberophile video is great but this one so nicely explains the things that we learn in school not even touching the point of the origin of SIN COS functions.
Math is so beautiful if learned in this way!
Ben is one of my favorite Numberphilers.
He's managed to explain 3 things I could not get my head around until now: the Mandelbrot set, the basics of trigonometry and The Golden Ratio to me using visual demonstration. I loves it.
I had a good maths teacher at school, don't get me wrong, (Thanks Mr. Noble) but I'm still quite jealous of the tools kids have now to better understand these concepts.
In school, we actually learned the definition of the trig functions in precisely this way and later applied them to triangles. Therefore I never quite understood how these functions are confusing to some. they are just beautiful. Furthermore, this approach is fantastic for learning about complex numbers and the R*(sin(phi) + i*cos(phi)) representation of complex numbers
Most schools do not put in the time or effort to teach them like this. It is truly unfortunate. Mathematical education in this world is at crisis.
Where were you 10 years ago?! 😭😭😭
All these years and finally i understand the circular functions! Thank you!
Love that sound effect where the equalization peak is going up and down the frequency range.
Is it just me, or does this chap look like William Riker from Star Trek?
just a bit?
The video does justice to the title absolutely beautiful man❤️
Cheers
I am taking finals this week at University and taking trigonometry. I have always been ok with math but this in depth trigonometry was tough. This video would have helped me so much this semester. It was so nice to see it visually. Wish I found you 12 weeks ago. Liked and subscribed now.
This is almost precisely the animation I had planned to create for my next music tutorial video (talking about harmonics and resonance). I need to get onto Geogebra by the looks! Thanks Ben and Brady (yet again) for a wonderful video.
The failure to actually mention the terms secant, cosecant, and cotangent while saying "sec", "cos", and "cot" is at best an odd choice. But notably makes life harder for those who might want to research them further.
Fair.
Not really, since the viewers most likely know what those abbreviations refer to already. This video is being made with the assumption that you already know what these functions are on a very basic level.
@@angelmendez-rivera351 not really... it is made with the assumption that you already have _used_ these functions, but without knowing on a very basic level what they are. The video in turn provides such understanding.
Silkwesir If you have used them, then you know what they are. It does not mean you have a deeper, insightful understanding of why they are what they are, but it still does mean you know what they are. Knowing what they are and understanding what they are constitute different things. Don't confuse the two, please.
You could just search for "sec trigonometry" and so on, instead.
This should be learnt in trigonometry firstly surely!
1:40 We actually use this circle (we call it "rotating vector") to find the phase of the oscilation and solve problems on simple harmonic motion in my physics class(12th grade physics)
Excellent visual representations and explanations!!
Wait, do people outside of the US actually call the tangent the "tan" and the secent the "sec?" When I was in school if we pronounced the abbreviations of tan, cos, etc we got yelled at.
I'm from Singapore, and during my math classes the teacher introduced it to us as "tangent, sine, and cosine functions", but from then on it was interchangeably used with "tan, sine, cos" since from context everyone knew what was being referred to.
It was always “tan of x” or whatever. Helped differentiate the tangent function from tangents.
Nope. Only for cos ,tan, cot ,we pronounce the abbreviations
I was taught that it's fine to say all of the abbreviations:
sin, cos, tan, sec, cosec, cot, shine, cosh, than, sesh, cosesh, coth
@@thehiddenninja3428 My calc professor usually said the full words for the trig functions of a circle, but he prudently decided it was too many syllables to say "hyperbolic sin", "hyperbolic cosine", etc. He always used abbreviations for those, pronounced (roughly) (IPA) /sintʃ/, /coʊʃ/, /tæntʃ/, /sitʃ/, /ˈkoʊˌsitʃ/, and /koʊθ/.
Imagine this stopped at 301 views
I haven' t seen such beautiful representation of these before. My life would have been incomplete without this. Thank you 😊
I have tried to learn trig so many times and I can never seem to wrap my mind around the trig functions. But this video really cleared it up for me. Thanks you so much!