Shine Bright: Solving the Equation \sinh 3x = 4\sinh^3x + 3 \sinh x

[chwala]

Homework Statement

Prove the hyperbolic function corresponding to the given trigonometric function.

$\sin 3x = 3\sin x- 4\sin^3x$

Relevant Equations

hyperbolic trig. properties.

We shall have;

$\sinh 3x = 3\sinh x- 4\sinh^3x$

$\sinh 3x =\sinh (2x+x)=\sinh 2x \cosh x + \cosh 2x \sinh x$

$\sinh 3x= 2\sinh x \cosh x \cosh x + (1+2 \sin^2x) \sinh x$

$\sinh 3x=2 \sinh x \cosh^2 x + \sinh x + 2\sinh^3x$

$\sinh 3x= 2\sinh x + 2\sinh^3 x + \sinh x + 2\sinh^3x$

$\sinh 3x = 4\sinh^3x + 3 \sinh x$

Insight welcome...today i have been singing shine shine shine

 

[pasmith]

See https://www.physicsforums.com/threa...-the-given-trigonometric.1047536/post-6824754.

 

[Mark44]

Homework Statement:: Prove the hyperbolic function corresponding to the given trigonometric function.

$\sin 3x = 3\sin x- 4\sin^3x$
Relevant Equations:: hyperbolic trig. properties.

We shall have;

$\sinh 3x = 3\sinh x- 4\sinh^3x$

$\sinh 3x =\sinh (2x+x)=\sinh 2x \cosh x + \cosh 2x \sinh x$

$\sinh 3x= 2\sinh x \cosh x \cosh x + (1+2 \sin^2x) \sinh x$

$\sinh 3x=2 \sinh x \cosh^2 x + \sinh x + 2\sinh^3x$

$\sinh 3x= 2\sinh x + 2\sinh^3 x + \sinh x + 2\sinh^3x$

$\sinh 3x = 4\sinh^3x + 3 \sinh x$

Insight welcome...today i have been singing shine shine shine

Rather than repeatedly writing the left-hand side in every equation, I find it easier to read (and write) as a chain of equalities leading from the original left-hand side to the final right-hand expression.

IOW, like this:
$\sinh 3x$
$=\sinh (2x+x)=\sinh 2x \cosh x + \cosh 2x \sinh x$
$=2\sinh x \cosh x \cosh x + (1+2 \sin^2x) \sinh x$
$= \dots$
$= 4\sinh^3x + 3 \sinh x$

 

[anuttarasammyak]

Homework Statement:: Prove the hyperbolic function corresponding to the given trigonometric function.

$\sin 3x = 3\sin x- 4\sin^3x$
Relevant Equations:: hyperbolic trig. properties.

Insight welcome...today i have been singing shine shine shine

Direct calculation from the formula
$$\sinh x=\frac{e^x-e^{-x}}{2}$$
$$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$
will give us the results including the different sign.

 

[Steve4Physics]

Homework Statement:: Prove the hyperbolic function corresponding to the given trigonometric function.

$\sin 3x = 3\sin x- 4\sin^3x$
Relevant Equations:: hyperbolic trig. properties.

We shall have;

$\sinh 3x = 3\sinh x- 4\sinh^3x$ ...

In case it's not already clear, what you tried to prove:
$\sinh 3x = 3\sinh x- 4\sinh^3x$
is incorrect. It should be:
$\sinh 3x = 3\sinh x+ 4\sinh^3x$
which is what you correctly proved,

Normal and hyperbolic trig' identities don't necessarily match. For example $cos^2x + sin^2x = 1$ but $cosh^2 – sinh^2x = 1$.

 

[robphy]

Using the method in the other thread: https://www.physicsforums.com/threa...n-trigonometric-function.1047536/post-6825226

$\begin{align*} \sin 3x &= 3\sin x- 4\sin^3x \\ (i\sinh 3u) &= 3(i\sinh u)- 4(i\sinh u)^3 \\ &\ \ \vdots \end{align*}$