Shatilar The periodic character of the trigonometric functions makes apparent that any point in their ranges is actually the image of infinitely many points. The derivatives of the hyperbolic functions follow the same rules as in calculus: Note that we often write sinh n x instead of the correct [sinh x ] nsimilarly for the other hyperbolic functions. To establish additional properties, it will be useful to express in the Cartesian form. Hyperbolic functions There is no zero point and no point of inflection, there are no local extrema. Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments.
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Welcome back to recitation. So this is, the subject is hyperbolic trig functions. So these are some interesting functions. But yeah, so let me introduce them and let me jump in just with their definition. So there are two most important ones. So for hyperbolic trig functions we have the hyperbolic cosine and the hyperbolic sine.
So the notation here, we write c o s h. So the h for hyperbolic. So hyperbolic cosine. And usually we pronounce this "cosh. And we usually pronounce this "sinch," so in American English as if there were an extra c in there.
So these functions have fairly simple definitions in terms of the exponential function, e to the x. So cosh of x is defined to be e to the x plus e to the minus x divided by 2.
And sinh of x is defined to be e to x minus e to the minus x divided by 2. So for cosh x, so we see as x gets big, so e to the minus x is going to 0. So it mostly is driven by this e to the x part.
And as x gets negative and big, then this is going to 0 and this is getting larger and larger. So we got something that looks like this. So it looks a little bit, in this picture it looks a little bit like a parabola, but the growth here is exponential at both sides. So in fact, this is growing much, much, much faster than, say, 1 plus x squared. And it reaches its minimum here at x equals it has the value 1 plus 1 over 2. So its minimum there is at x equals 0, it has its minimum value 1.
So we got exponential growth off that side. So as x goes to minus infinity, this curve goes also to minus infinity. And again, the growth here is exponential in both cases. So this is a sort of basic picture of what these curves look like.
They have some nice properties, and let me talk about them. Exponential functions are easy to take the derivatives. Take the derivative of e to the x, you get e to the x. So the derivative of cosh x is e to the x minus e to the minus x over 2. But we have a name for this. This is actually just sinh x. So the derivative of cosh is sinh, and the derivative of sinh, well, OK.
You look at the same thing, take this formula, take its derivative. Well, e to the x, take its derivative, you get e to the x.
So this is e to the x plus e to the minus x over 2, which is cosh x. For trig functions, if you take the derivative of sine you get cosine. And if you take the derivative of cosine you almost get back sine, but you get minus sine. So you, when you take the derivative of cosh you get sinh on the nose. No minus sign needed. But the real reason that these have the words trig in their name is actually a little bit deeper.
So let me come over here and draw a couple pictures. So the unit circle has equation x squared plus y squared equals 1. Well, close enough, right? And what is the nice relationship between this circle and the trig functions?
Well, if you choose any point on this circle, then there exists some value of t such that this point has coordinates cosine t comma sine t. Now it happens that the value of t is actually the angle that that radius makes with the positive axis.
But not going to worry about that right now. So as t varies through the real numbers, the point cosine t, sine t, that varies and it just goes around this curve.
So it traces out this circle exactly. So the hyperbolic trig functions show up in a very similar situation. But instead of looking at the unit circle, what we want to look at is the unit rectangular hyperbola. So what do I mean by that?
So this is the equation x squared minus y squared equals 1. So this is the graph of the equation x squared minus y squared equals 1. Now what I claim is that cosh and sinh have the same relationship to this hyperbola as cosine and sine have to the circle. So x squared minus y squared.
So this is, so we use most of the same notations for hyperbolic trig functions that we do for regular trig functions. So this is cosh squared u minus sinh squared u. And now we can plug in the formulas for cosh and sinh that we have.
So this is equal to e to the u plus e to the minus u over 2, quantity squared, minus e to the u minus e to the minus u over 2, quantity squared.
And now we can expand out both of these factors and-- both of these squares, rather, and put them together. So over 2 squared is over 4 and we square this and we get e to the 2u. OK, so then we get 2 times e to the u times e to the minus u.
But e to the u times e to the minus u is just 1, so plus 2. Plus e to the minus 2u minus e to the 2u minus 2 plus e to the minus 2u-- so same thing over here-- over 4. So this is 4 over 4, so this is equal to 1.
So if x is equal to cosh u and y is equal to sinh u, then x squared minus y squared is equal to 1. So if we choose a point cosh u, sinh u for some u, that point lies on this hyperbola. That this point-- OK, so the point cosh u, sinh u is somewhere on this hyperbola.
If you look at all such points, if you let u vary and look-- through the real numbers and you ask what happens to this point cosh u, sinh u , the answers is that it traces out the right half of this hyperbola. So let me say one more thing about them, which is that we saw that they have this analogy with regular trig functions. So instead of satisfying cosine squared plus sine squared equals 1, they satisfy cosh squared minus sinh squared equals 1.
And instead of satisfying the derivative of sine equals cosine and the derivative of cosine equals minus sine, they satisfy derivative of cosh equals sinh and derivative of sinh equals cosh. So similar relationships. Not exactly the same, but similar. So one example of such a formula is your-- for example, your angle addition formulas. So, exercise. Find sinh of x plus y and cosh of x plus y in terms of sinh x, cosh x, sinh y, and cosh y.
So in other words, find the corresponding formula to the angle addition formula in that case of the hyperbolic trig functions. Free Downloads.
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