Hyperbolic functions Definition and 80 Threads

I was able to solve with a rather longer way; there could be a more straightforward approach; My steps are along these lines; ##\sinh^{-1} x = 2 \ln (2+ \sqrt{3})## ##\sinh^{-1} x = \ln (7+ 4\sqrt{3})## ##x = \sinh[ \ln (7+ 4\sqrt{3})]## ##x = \dfrac {e^{\ln (7+ 4 \sqrt{3})} - e^{-[\ln 7+ 4...

My question is on the highlighted part (circled in red); Why is it wrong to pre-multiply each term by ##e^x##? to realize , ##5e^{2x} -2-9e^x=0## as opposed to factorising by ##e^{-x} ## ? The other steps to required solution ##x=\ln 2## is quite clear and straightforward to me.

I have, Using ##\ cosh 2x = 2 \cosh^2 x - 1## ##\cosh x = 2 \cosh^2\dfrac{x}{2} -1## Therefore, ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -1 - 1## ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -2## ##=2\left[ \cosh^2 \dfrac{x}{2}...

In my approach i have the following lines ##\ln (x + \sqrt{x^2+1}) = 2\ln (2+\sqrt 3)## ##\ln (x + \sqrt{x^2+1} = \ln (2+\sqrt 3)^2## ##⇒x+ \sqrt{x^2+1} =(2+\sqrt 3)^2## ##\sqrt{x^2+1}=-x +7+4\sqrt{3}## ##x^2+1 = x^2-14x-8\sqrt 3 x + 56\sqrt 3 +97## ##1 = -14x-8\sqrt 3 x + 56\sqrt 3 +97##...

The eqution can be written as: ## Eq= e^{t( -h + \sqrt{ x} )} + e^{t( -h -\sqrt{ x} )} ## Can this be written in terms of Cosh x ?

I found one ages ago about the hyperbolic functions, but it hadn’t been translated to English from German yet. Anyone know of a good book on hyperbolic functions and other transcendental functions besides the circular functions (trigonometric)?

My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...

##8 \sin^4u = 3-4\cos 2u+\cos 4u## ##8 \sinh^4u = 3-4(1+2\sinh^2 u)+ \cosh ( 2u+2u)## ##8 \sin^4u = 3-4-8\sinh^2 u+ \cosh 2u \cosh 2u + \sinh 2u \sinh 2u## ##8 \sinh^4u = 3-4+1-8\sinh^2 u+ 4\sinh^2u +4\sinh^4 u + 4\sinh^2 u + 4\sinh^4 u## ##8 \sinh^4u = -8\sinh^2 u+ 8\sinh^2u +8\sinh^4 u##...

This is a textbook question and i have no solution. My attempt: We know that ##\cosh x = \dfrac{e^x + e^{-x}}{2}## and ##\cosh u = \dfrac{{x^2 + 1}}{2x}## it therefore follows that; ##e^{2u} = x^2## ##⇒u = \dfrac {2\ln...

I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties... My approach on this; ##\tanh x = \sinh y## ...just follows from ##y=\sin^{-1}(\tan x)## ##\tan x = \sin y## Therefore...

While deriving Lorentz transformation equations, my professor assumes the following: As ##\beta \rightarrow 1,## $$-c^2t^2 + x^2 = k$$ approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...

Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...

The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are $$ \begin{cases} \sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x)...

Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...

I am trying to understand why maxwell equations are correct in any reference frames? While i started to understand of his laws of physics a bit i could not imagine why he uses hyperbolic functions such as coshw instead of spherical ones in position and time relation between moving frames...

Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...

Mod note: Because his caps-lock key is stuck, it's OK for this post to be in all caps. FIRSTLY, MY LAPTOP'S CAPS LOCK IS BEHAVING REALLY WEIRD AND I HAVE NO CONTROL ON IT WHATSOEVER. SO SORRY FOR POSTING IN ALL CAPS/ALL SMALL LETTERS I HAVE RECENTLY LEARNED HYPERBOLIC FUNCTIONS. HOWEVER, I AM...

Homework Statement Can this function be integrated analytically? ##f=\exp \left(-\frac{e^{-2 \theta } \left(a \left(b^2 \left(e^{2 \theta }-1\right)^2 L^2+16\right)-32 \sqrt{a} e^{\theta }+16 e^{2 \theta }\right)}{b L^4}\right),## where ##a##, ##b## and ##L## are some real positive...

Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...

Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...

I noticed the graphs of ##y=\cos^{-1}x## and ##y=\cosh^{-1}x## are similar in the sense that the real part of one is the imaginary part of the other. This is true except when ##x<-1## where the imaginary part of ##y=\cos^{-1}x## is negative but the real part of ##y=\cosh^{-1}x## is positive. I...

Homework Statement find the limit of arccoshx - ln x as x -> infinity Homework Equations ##arccosh x = \ln (x +\sqrt[]{x^2-1} )## The Attempt at a Solution ## \lim_{x \to \infty }(\ln (x + \sqrt{x^2-1} ) - \ln (x)) = \lim_{x \to \infty} \ln (\frac{x+\sqrt{x^2-1}}{x}) \ln (1 + \lim_{x \to...

Homework Statement Split the function f(x) = ex + πe−x into odd and even parts, and express your result in terms of cosh x and sinh x. Homework Equations f(x) = 0.5[f(x) + f(-x)] +0.5[f(x) - f(-x)] The Attempt at a Solution So i know that: ex = 0.5[ex - e-x] + 0.5[ex + e-x] = sinh(x) +...

Good afternoon, i was just wondering if this equation is possibly solvable where I(z) is a function of z. The equation is: I(z)=cosh(1/2 ∫I(z)dz) I know it looks stupid but is it possible? How would you approach this problem? Thank you.

1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...

What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.

To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...

More than just a few problems that happen to pop up in the textbook, I mean.

Is known that in every rectangle triangle the following relationships are true: But, how use geometrically the function sinh, cosh, and tanh?

Homework Statement Attached is the problem Homework Equations My question is am i going about it the right way for question C). I have done A and B and am sure they are correct. The Attempt at a Solution Attached

Homework Statement I thought it would be better to attach it. Homework Equations The Attempt at a Solution So for the first part I've found that A^2=the Identity matrix, but from there I don't have much of an idea on how to substitute that into the equation for M and end up with...

show that the locus of the point \left(\dfrac{a(cosh\theta + 1)}{2cosh\theta},\dfrac{b(cosh\theta - 1)}{2sinh\theta}\right) has equation x(4y^2 + b^2) = ab^2 working: http://gyazo.com/4c96af128d0293bce7f18029c2f54b0d where have I gone wrong :(

[SOLVED] Fourier Series Involving Hyperbolic Functions Hello everyone! Sorry if this isn't the appropriate board, but I couldn't think of which board would be more appropriate. I was running through some problems I have to do as practice for a test and I got stuck on one I'm 99% sure they'll...

Will someone help me to understand sinh, cosh, and tanh. I know they have some relevance to hyperbolas and trigonometric identities. Thank you.

I've searched and thought on it for a long time but I couldn't find any mathematical proof or something else about the formula of hyperbolic functions. sinh=\frac{e^{x}-e^{-x}}{2},cosh=\frac{e^{x}+e^{-x}}{2} How do I get these formulas mathematically??

Homework Statement Prove that cosh (\frac{x}{2}) = \sqrt{\frac{cosh(x)+1}{2}} Homework Equations cosh(x) = \frac{e^{x}+e^{-x}}{2} The Attempt at a Solution \frac{\sqrt{e^{x}}+\sqrt{e^{-x}}}{2} \ast \frac{\sqrt{e^{x}}-\sqrt{e^{-x}}}{\sqrt{e^{x}}-\sqrt{e^{-x}}} \rightarrow...

Hi, I am trying to integrate (tanh(x)+coth(x))/((cosh(x))^2) I am substituting u=tanh(x), du=dx/((cosh(x))^2) and end up with 1/2(tanh(x))^2 + ln |tanh(x)| + C which is incorrect. What am I doing wrong??

Homework Statement Compute the following: \int \frac{cosh(x)}{cosh^2(x) - 1}\,dx Homework Equations \int cosh(x)\,dx = sinh(x) + C The Attempt at a Solution I had no clue where to start, so I went to WolfRamAlpha, and it used substitution but it made u = tanh(\frac{x}{2})...

Homework Statement Prove sin(x-iy) = sin(x) cosh(y) - i cos(x) sinh(y) Homework Equations The Attempt at a Solution I tried to prove it by developing sinh into it's exponential form, but I get stuck. sinh(x-iy) = [ ei(x-iy) - e-i(x-iy) ] /2i = [ eixey - e-ix e-y ] /2i...

Homework Statement I am a little confused on the steps to take to solve these kinds of functions. Solve: cosh z = 2i The Attempt at a Solution We were given identities for sinh z = 0 and cosh z = 0 and also other identities like cosh(z) = cos (iz) So cos (iz) = 2i cos...

Homework Statement Find \int \frac{x}{\sqrt{2x^2-2x+1}}\,dx The attempt at a solution First, i complete the square for the quadratic expression: 2x^2-2x+1=2((x-\frac{1}{2})^2+\frac{1}{4}) \int \frac{x}{\sqrt{2x^2-2x+1}}\,dx=\int \frac{x}{\sqrt 2 \sqrt{(x-\frac{1}{2})^2+\frac{1}{4}}}\,dx...

Bany's question from Yahoo Questions: CB

Homework Statement why is the answer not e^10x ? If you ignore the e's it should be 5x - 5x + 5x - - 5x, or 5x - 5x + 5x + 5x, which is 10x

Homework Statement Express the function cosh(6x) in terms of powers of cosh(x) Homework Equations The Attempt at a Solution Okay the problem booklet also asks me to do the opposite. Express cosh(x)^6 as mutiples of cosh(x). I can do that fine, I just simply write it out as [1/2(e^x + e^-x)]^6...

Hi, My brain is not working today. So can someone please tell me what I am doing wrong. (^2 = squared) coshy^2 - sinhy^2 = 1, how do I rearrange this for coshy^2 I keep getting: coshy^2 = 1 + Sinhy^2 The book that I'm looking at has it this way: coshy^2 = Sinhy^2 + 1 Thanks Obs

Where do the definitions of hyperbolic functions in terms of exponentials come from ?

Homework Statement \int cosh(2x)sinh^{2}(2x)dx Homework Equations Not sure The Attempt at a Solution This was an example problem in the book and was curious how they got to the following answer: \int cosh(2x)sinh^{2}(2x)dx = \frac{1}{2}\int sinh^{2}(2x)2cosh(2x) dx =...

Homework Statement The area bounded by y=2 coshx, the x-axis, the y-axis, and the line x=4 is revolved about the x-axis. Find the volume of the solid generated. Homework Equations I sliced the area along the axis of revolution. That is the strip is dx. So the equation necessary is...

Hyperbolic functions :((( Homework Statement Question is: and Homework Equations Now from what i can recall the formula for sin(A+B) = sinAcosB+sinbcosA so same goes for hyperbolic function i suppose? sinh(2x+x) = sinh2xcoshx+cosh2xsinx (2sinhxcoshx)coshx +...

1. Differentiate cosh(x) using first principles 2. cosh(x) = (e^x+e^-x)/2 From previous exercises, I know the answer will be sinh(x)= (e^x-e^-x)/2 but I cannot get to the answer. I seem to be left with the equation: lim h ---> 0 (e^2x*e^2h +1-e^h*2e^x +e^h)/(2h*e^x*e^h) But when...