Hyperbolic functions and the unit hyperbola | Hyperbolic functions | Precalculus | Khan Academy
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Hyperbolic functions and the unit hyperbola | Hyperbolic functions | Precalculus | Khan Academy


We know that if we take all of the points in the *X, Y planewherex^2 + y^2=1*, we get ourselves the unit circle. Let me draw the unit circle. That’s myy-axis; this is myx-axis. And the unit circle has the circle with the radius one. So that’sx=1, that’sx=-1, that’sy=1, that’sy=-1the unit circle looks something… let me draw it… something like this, I think you get the point Let’s see if I can fill it in a little bit better. So you realize that it’s not a dotted circle. There. That’s my best attempt at drawing the unit circle. And we also know that the traditional trig functions, or maybe we should call them thecirculartrig functions are actually defined so that if you parameterize so if you were to take *x=cos t* and *y=sin t* and you pick anyt, right over here and by definition it’s going to sit on the unit circle by definition, *x^2 + y^2=1* so if you pick anytit’s going to sit some place on this unit circle. Or another way to think of it is if you varytit’s going to start tracing out this circle And we know thattcorresponds to the angle with a positivex-axis, in this case, that right over there ist. Now wouldn’t it be neat if there were a similar analogy for, not the unit circle, but something we could call the unit hyperbola? So that’s our little review of trigonometry right there; our traditional trigonometry, now let’s think about the unithyperbola. Well, *x^2 + y^2=1* is a unit circle, I’ll say that *x^2 – y^2=1, I’m going to call this myunit hyperbola*. Or a unit rectangular hyperbola.Hyperbola. This is just a little bit of review from *Conic Sections*, but it would look something like this: It would look… something… that’s myy-axis, this is myx-axis, and then we can say, well ifyis 0,xcan be ±1, so you can think of that as the unit part where it intersects thex-axis; that’s +1, that’s -1 and it has asymptotes,y=xandy=-xWe go through the intuition on that in the Conic Section videos,y=xis that dotted line,y=-xis that dotted line, right over there, and then this thing is going to look like this. It’s going to have a right half that does something like this, and does something like this, all a review of *Conic Sections*, it gets closer to its asymptotes. Toy=xory=-xand the same thing on the left-hand side. It’s going to do something like that. Wouldn’t it be neat if we could parameterizexandywith analogous functions so that we get a similar type of property? And you might guess what those functions are, but let’s actually try to verify it. What would happen ifxis equal to ourhyperbolic*cosine of t*, which is the same thing as *e^t + e^(-t)*, all of that over 2 andywere to be equal to *hyperbolic sine* oft, which is equal to *e^t – e^(-t)* over 2. Wouldn’t it be neat if there were an analogy here; over here you pick anytbased on our circular trig functions, you ended up with a point on the unit circle. Wouldn’t it be amazing if for any pointtyou ended up with a point on our, what we’re calling our *unit hyperbola*? Well, in order for that to be true, with this parameterization *x^2 – y^2* would need to be equal to one. Let’s see if thatisthe case! So *x^2 – y^2* is equal to, well let’s square this business it’s equal toe^(2t)plus two times the product of these two things *2e^t • e^(-t)*, this ise^0here which is 1. Pluse^(-2t),e^(-t)^2, all of that over 4 And then from that we will subtracty^2. Minus, so the numerator’s going to be *e^(2t) – 2e^t • e^(-t) + e^(-2t)*, all of that over 4 So, immediately, a couple of simplifications here. *e^t • e^(-t)*, that’s juste^(t-t)which is equal toe^0, which is equal to 1 This is going to be one, that’s going to be one, so we’re going to have a 2 in either of those cases and if we were to simplify it, all of this stuff over here I’ll do a numerator, so this is going to be equal to over our [denominator] of 4 *e^(2t) + 2 + e^(-2t) – e^(2t)* just distributing the negative sign Plus two, and then minuse^(-2t)Well this is convenient! (Oh, I was writing it in black, a hard color to see) This cancels with this, This and this also add up to zero and you’re left with two plus two over four which is indeed equal to one! So this is a pretty good reason to call these two functions hyperbolic trig functions. These are the circular trig functions, you give me aton these parameterizations we end up on the unit circle! You varyt, you trace out the unit circle. Here, for any realt, we’re going to assume we’re dealing with real numbers, for any realtwe’re going to end up on the unit hyperbola right over here and in particular we’re going to end up on the right so it’s not exactly… over here pretty muchanyof these points could be parameterized right here over here we’re going to end up on a point on therightside of the unit hyperbola. The reason why it’s the right side is… you go straight to the definition of *cosh t*, this thing can only be positive This thing can only be positive.e^tcan only be positive,e^-tcan only be positive so this is only positive. But you give anytyou will end up on this hyperbola! Specifically the right side, if you want points on the left hand side, you’d have to take the *-cosh tand thesinh t* to end up right over there. But it’s a pretty neat analogy. We’re looking at Euler’s identity and we kind of said, “oh, let’s just start playing with these things!” There seems to be a similarity here if we were to remove thei‘s and, all of a sudden, we’ve discovered another thing! That there is this relationship here there is this relationship betweenthesetrig functions and the unit circle, here between ournewlydefined hyperbolic trig functions and the unithyperbola. And you’d also find if you were to varytit’s going to trace out… just as if you were to varythere it traces out the unit circle… if you tracethere it will trace out the right-hand side, the right-hand side of the unit hyperbola. For this parameterization right here.

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72 thoughts on “Hyperbolic functions and the unit hyperbola | Hyperbolic functions | Precalculus | Khan Academy

  1. than how come the cosh and sinh functions are defined for angles above 45 degrees but below 135 degrees? lines that go through (0,0) that intersect the X axis at those angles never intersect with the unit hyperbula… I understand that the exponential functions are still defined, for any t, but isn't that a big hole in the analogy?

  2. It's something called the "hyperbolic angle". It's a bit tricky to explain, but it would be sweet if Sal did a video on it.

  3. Do not think of mathematics as 'useful.' That mindset is the reason why the education system is so horrid.

  4. I don't the guy meant it like (not really his fault i blame teachers, in general), but totally agree with your distaste for the education system.

  5. I just want to point out something. The reason for this underlying awesome connection with cos, sin, cosh, and sinh, is because if you extend a unit circle into the imaginary plane, it traces out a hyperbola. Euler's Identity works with this because if you use an imaginary angle, e^(i(-ix))=cos(ix)-isin(ix). From this, you can derive that cos(ix)=cosh(x), and -i*sin(ix)=sinh(x). Just another neat connection that fills in some gaps between discoveries.

  6. yeah you are totally right. if I set x=f(t)=2 and y=g(t)=1 I also fulfil x^2-y^2=1 for all t, but it obv doesnt spread over the entire parabola, its just one point.

  7. Totally this. I'm desperately trying to uncover the connection between boosts and hyperbolic trig functions…

  8. Okay, so on the unit circle "t" is an angle rising counter-clockwise from the x-axis to hit a point on the unit circle. Is it the same for hyperbolic trigonometric functions? In that case, I'd expect 45° or π/4 to be infinity.

  9. You are awesome at drawing with the mouse
    but i still didn't undertstand why did they even come up with these functions???
    how did they write them in terms of e^(x) amd all that ???

  10. You are awesome at drawing with the mouse
    but i still didn't undertstand why did they even come up with these functions???
    how did they write them in terms of e^(x) amd all that ???

  11. I am in first year of physics at university and at math lesson the teacher didnt explain anything.But YOU ARE REALLY AWSOME!!!Thank you for saving me!!!Greetings from Greece!!!!

  12. you didn't explain a thing its like you consider us know already what hyperbola unit is you just start out without introduction and you made it more scarier

  13. wow!!! you do not realise the the light bulb that just turned on nice and bright after this video!! thank you so much explains sooooo many things i missed.!!!

  14. For circle, t is ankle and is represented. For hyperbole, give geometrical interpretation for t. Nobody gives the interpretation for this.

  15. sweet o/ with a tiny little bit of help from khan explaining what the hell a cosh is in mathemetical form I found myself a way to derive Cos(x)Cosh(y) – Sin(x)Sinh(y) = x -iy = z.
    So now I'm going to celebrate and feel like a mathgod for a second before I continue on with my assignment xD

  16. What seems to be lacking in math tutorials is the background history to the origins of the concepts , mathematicians, it only need to be brief statement to link how math evolved into the list of axioms

  17. im 17 with major scientific and mathematical curiosity, yet i cant understand basic math but im so desperate to, i wanna be able to self study and teach myself geometry and other hyperbolic related stuff if that makes sense. i dont know where to start, ive always been in the lowest grade of math possible in highschool because i simply cant conceptualise it, but i know there's a way for me to understand the concept, if i went straight into tafe or uni to learn this, i wouldnt have a clue, and i need a proper start of how to understand the most fundamental basics of basic stuff as possible. i really need help with this. i'm much more comfortable being self taught, as schooling systems give u a due date to learn and understand this stuff, which i feel is either too quick for my pace, or they try to teach and cram everything so dense i cant grasp the fundamental basics of it, i need self teaching help in order for me to learn this on my own so then i can then go forward with an education that does have due dates and fast pace learning, so then i can atleast understand whats going on in class and feel confident about a potential future. please if anything, i'd love for people to suggest me the titles of such basics for me to learn and then move forward onto. please. i really need this. i love science with such a hard passion, but i cant understand or learn any of it due to math being such an involvement in science. i really need help. please. thank you.

  18. I like "hyperbolic cosine theta" equals the quantity e raised to the i theta plus e raised to the minus i theta the quantity divided by two from Khan Academy and I have a graphical proof.

  19. oh hell this is how you explain hyperbolic functions college teacher ruined it with just formula writing and solving problems based on formula that's all she did

  20. I was getting it and then you did something that all my previous teachers have done and lost me. You started working ou x2 – y2 = e 2t + 2et e-t, without explaining where the additonal Es come from and why. Then it became over 4 and not 2? (5minutes into Video you started to talk quickly and assume i know what your on about) haha my head hurts

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