Creating a function similar to sin(x) with steeper downturns but equal peaks and lows As the title states, I've been trying to create a function that is identical to $\sin(x)$, except that it has a steeper slope in the middle of the decreasing portion of the curve, but mellows out such that this function and $\sin(x)$ have the same turning point for both peak and bottom.
I've tried to describe it in the following image, if we pretend the top function is $\sin(x)$, I'd like to create something akin to the bottom image. Currently I am trying some variation of $\sin(x-0.5\sin(x))$, which gives me the steeper slope, but this shortens the actual duration and the peaks and lows don't line up.
Any hints on functional forms I could try would be appreciated!
Pardon my extremely poor paint skills:

 A: What do you mean when you say "create"?   You can take $H\circ  \sin(x)$, where H has the properties:  $H(x)$  close to $-1$  for $x \le -1/3$,  $H(x)$ close to  $1$ for $x \ge 1/3$,  $H$ is weakly increasing,  $H$ is smooth, and $H(-x) = -H(x)$  You can "create" such an $H$ by taking a piecewise linear function and then convolving with a smoothing function (e.g. normalized Gaussian).  
I programmed this in Mathematica. Here's the plot of $H$, followed by the plot of $H\circ \sin$.


Code:
h[x_] := Piecewise[{{-1, x <= -1/3}, {1, x >= 1/3}}, 3 x ]
A = 1/4;    (*parameter for Gaussian;  smaller gives narrower peak *)
G[x_] :=  Exp[-(x/A)^2/2]/(A Sqrt[2 Pi]); 
H[y_] = Convolve[h[x], G[x], x, y]; 
L[x_] :=  H[Sin [x]]
Plot[h[x], {x, -2, 2}]
Plot[H[x], {x, -2, 2}]
Plot[L[x],  {x, -10, 10}]

A: Here's a fairly straightforward solution:
$ f_k(x) = \arctan(k \cdot \sin x) / \arctan k $
Steep sine
We are using a non-uniform scaling function
$ f_k(x) = \arctan(k \cdot x) / \arctan k $
where $k$ determines the degree of deformation:

*

*with small $k$ (e.g. $0.001$) it's almost identity transform.

*with larger $k$ (e.g. $4$ or more) there's more upward/downward deformation

Because the domain is the result of $\sin x$, we don't worry what happens outside $[-1..1]$, so we don't need to make it piecewise.
Steep deformation
Other non-uniform scaling functions in the Sigmoid function family are applicable.
An application in image processing is for producing high contrast transformation.
