For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.

learn more… | top users | synonyms

0
votes
1answer
24 views

Solution of $A = e^{\alpha t}\cos(\omega t + \phi)$

I would like to find the real roots of the function $$i(t) = \frac{\hat{V}}{R}\left(\frac{\omega^2}{(\alpha^2 + \omega^2)} \cos\left(\omega t + \tan^{-1}\left(\frac{\alpha}{\omega}\right)\right) + ...
2
votes
2answers
16 views

Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.
0
votes
2answers
21 views

Finding the values of hyperbolic functions

If sinh x = 4/3 , find the values of the other hyperbolic functions at x.
2
votes
2answers
59 views

Simplify $\sinh (\log (x))$

$$\sinh (\log (x))=\frac{x^2-1}{2 x}$$ However I do not see how this is done, here is an idea I had but I'm probably way off: $$\sinh \left(\ln \left(\frac{1}{2} ...
0
votes
0answers
21 views

hyperbolic series $\sum_{r=1}^n \cosh(rx)$

I have attempted to do this question on hyperbolic functions: Prove that $$\cosh x + \cosh 2x + ... + \cosh nx ...
4
votes
3answers
87 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...
0
votes
2answers
23 views

Tangent - point of contact

Question: Tangent to the curve $y = x^2 + 6$ at point P(1, 7) touches the circle $x^2 + y^2 + 16x + 12y + c = 0$ at a point Q. Then the coordinates of Q are: 1) (-6, -11) 2) (-9, -13) 3) (-10, -15) 4) ...
0
votes
2answers
53 views

A Method For Calculating Large Exponents Quickly

I've derived a formula for calculating large exponents quickly: $$a^b = 2 \cosh( - b \log( a ) )$$ My question is: Has anyone seen anything similar? I am curious if either it's novel OR if I have ...
0
votes
0answers
43 views

Hyperbolic sinc function

Cardinal sine function or sinc function is defined by: \begin{equation} \mathrm{sinc}x=\begin{cases}\frac{\sin x}{x}, & x \neq 0,\\ 1, & x = 0,\end{cases} \end{equation} Is there any ...
1
vote
2answers
29 views

here is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

There is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$. My question is, isn't $z \rightarrow kz$ an isometry for ...
0
votes
0answers
6 views

How can I find a hyperbolic function denoting zoom levels?

I'm working between two values. The first ($m$) represents the number of meters wide an estimate of location accuracy is, and the other ($z$) represents a vague level of zoom as described below. This ...
2
votes
0answers
23 views

Proof that function is real part of $\sec(z)$ [duplicate]

I'm working on the following problem: I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the ...
0
votes
1answer
37 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
0
votes
3answers
68 views

How does $\frac{1}{2}\cosh(2x) -1 = \sinh^2(x)$?

Using hyperbolic trigonometric function identities is there a way to prove the following equation? $$\frac{1}{2} (\cosh(2x)-1) = \sinh^2(x)$$
3
votes
2answers
423 views

Calculate cosh(x) given sinh(x)

Given the value of sinh(x) for example sinh(x) = 3/2 How can I calculate the value of cosh(x) ?
4
votes
4answers
217 views

What is the use of hyperbolic trigonometric functions if they are easily expressible algebraically?

I get that there are uses for $\sin(x)$ and $\cos(x)$ because they are defined with imaginary exponents which aren't as easily worked with but the hyperbolic functions are simply $\frac12(e^x\pm ...
1
vote
1answer
17 views

rotated hyperbolic cylinder parameterization

A hyperbolic cylinder is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1$, but thats a hyperbolic cylinder that goes along the Z-axis. How do you parametrize a hyperbolic cylinder that goes instead of ...
0
votes
0answers
60 views

Mathematica Integrate gives back the integrand

i'm trying to Integrate the following function: (q (1 + q) - E^-q Sinh[q])/(-q + Cosh[q] Sinh[q]) - ( 2 q Tanh[q])/(-q + Cosh[q] Sinh[q]) I already solved ...
4
votes
5answers
179 views

Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$.

Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$ I am having some trouble proving this without derivative. Some help would be much appreciate!
0
votes
0answers
52 views

Exact Values of Hyperbolic Trig Functions

There are some well-known exact values for trig functions, such as $$\sin\frac{\pi}{6}=\frac{1}{2},\quad \tan\frac{\pi}{3}=\sqrt 3, \quad\text{etc.}$$ Are there comparable special values for the ...
1
vote
2answers
113 views

Hyperbolic function identity proof?

On a question i am working thru it says: Obtain the formula:$$ \sinh 2x - \sinh 2y = 2\cosh(x+y)\sinh(x-y) $$and prove that $$\coshθ + \cosh2θ +...+\cosh nθ ...
0
votes
0answers
39 views

Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
1
vote
1answer
40 views

How to trace the graphic of $\cos(x) + \cosh(y) = k$?

Is there some systematic way to trace the graphic of $\cos(x) + \cosh(y) = k$ given a fixed value for $k$? Suppose $k = 1$: if I choose empirically $y = 1.2$, I know that should be $\cos(x) = - ...
0
votes
1answer
42 views

Exponential function to prove [closed]

how would you prove that $Ae^x+Be^{-x}=A \sinh x+B\cosh x$ Thank you.
5
votes
5answers
202 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
3
votes
2answers
77 views

Why are hyperbolic functions significant?

I'm currently covering Stewart's Early Transcendentals, and there is a whole section dedicated to defining and differentiating hyperbolic functions. The same amount of space is used to cover other ...
0
votes
1answer
97 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
0
votes
1answer
51 views

Hyperbolic equations-characteristic system

Let the system: $$\alpha(x,t,u)u_t+\beta(x,t,u)u_x=f(x,t,u)$$ To find the characteristic equations: $$\frac{du}{ds}=\frac{\partial{u}}{\partial{t}} \frac{dt}{ds}+\frac{\partial{u}}{\partial{t}} ...
0
votes
2answers
40 views

Find $\sinh x$ in terms of $\tanh x$.

Given that $\tanh(x) = u$, find an expression for $\sinh(x)$ in terms of $u$. I don't really know what the question wants from me here. Any help would be great.
0
votes
2answers
20 views

Find the normalisation constant

I am having problems finding the normalisation constant $N$. I have tried this so far use the substitution $x=a tan(u)$ so $dx=a sec^2(u)du$, so $\displaystyle 1=\int_{-\infty}^{\infty}N^2 ...
0
votes
1answer
43 views

Hyperbolic growth, deriving from hyperbolic functions

When a quantity grows towards infinity in a finite-time, it is said to undergo hyperbolic growth. An example being a quantity that every time it doubles, the growth rate itself also doubles. ...
1
vote
1answer
52 views

Infinite Product Identity for Hyperbolic Sine

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$. I saw this formula in a book and have no idea ...
4
votes
0answers
40 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
0
votes
1answer
27 views

showing $\sinh(x/2) = \epsilon \sqrt{\frac{1}{2}(\cosh(x) -1)}$

showing $$\sinh(x/2) = \epsilon \sqrt{\frac{1}{2}(\cosh(x) -1)}$$ and I was told to determine the value of $\epsilon$. From identities I reached $ \sinh^2(x) = \dfrac{1}{2}(\cosh(x) -1)$ however ...
3
votes
1answer
75 views

given $\cosh u = x$ find $\sinh u$

I'm asked to show that:$\newcommand{\arcosh}{\operatorname{arcosh}}$ $\int{x \arcosh x}dx = \frac{1}{4}(2x^2 -1)\arcosh x - \frac{1}{4}x\sqrt{x^2 -1} + C$ If I integrate by parts: let $u = \arcosh ...
1
vote
1answer
57 views

Limit of a Cosh function

Evaluate $$\lim_{t\to\infty} (\cosh x)^{1/x}.$$ I tried to use L'Hopital's but I think I made a mess of the differentiation, and the differentiation doesn't seem like it'll help much.
1
vote
1answer
68 views

Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
1
vote
3answers
123 views

Definition of hyperbolic cosine and its relation with exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
1
vote
1answer
37 views

Disappearing negative signs when evaluating a sinh^-1 integral

$$\int_{-2}^{6} \frac{1}{\sqrt{1+(-x)^2}} \, dx$$ When performing this integral on paper, I get $$\sinh^{-1}(6) - \sinh^{-1}(-2) $$ But when I type it on wolframalpha, I get the unintuitive answer ...
1
vote
2answers
40 views

Complex Numbers and Hyperbolic Functions

How would you evaluate: $\mathfrak{R}\left[(1+i)\sin\left(\dfrac{(2+i)\pi}{4}\right)\right]$? I know that $\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$ and $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$. I have also ...
2
votes
1answer
44 views

Inverse trig and trigh in integration?

I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is ...
1
vote
3answers
406 views

Separate Into Real and Imaginary Parts

Separate the following trigonometric function into Real and Imaginary Parts $$\tan^{-1}e^{i\theta} $$ or $$\tan^{-1}(\cos\theta+i\sin\theta)$$ I Have made till here Assuming $x+iy$ is the final ...
0
votes
1answer
48 views

Find the value of hyperbolic $\tanh x$ function from the equation

If $\sinh x-\cosh x=5$, find $\tanh x$ I have done till the following steps but dont know how to proceed further from solving this equation in Euler's form ...
1
vote
2answers
44 views

Does the definite integral of (1 - tanh t) from 0 to x diverge as x goes to infinity?

Decidedly in the category of things I used to know how to prove but have forgotten: Does $$ \int_0^x (1 - \tanh t) \,dt $$ converge or diverge as $x \to \infty$? (I know that the indefinite ...
0
votes
1answer
41 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
0
votes
2answers
47 views

Is this a typo, or am I missing something?

I have a handout for my precalc II class. It says $\sinh(-x) = -\sin(x)$ It should be $\sinh(-x) = -\sinh(x)$ right? I don't see how a negative input could make a hyperbolic function circular.
1
vote
1answer
65 views

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

I'm solving the Laplace equation $U_{xx}+U_{yy}=0$ subject to BC's: \begin{align} U(0,y) &= 0 \\ U(a,y) &= 0 \\ U(x,0) &= 0 \\ U(x,b) & = \left\{x \text{ for } x \in ...
1
vote
2answers
27 views

Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
0
votes
2answers
86 views

I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ...