# (sin )^(-1) is the piecewise-wholesome inverse sine operator. The FCS y = (sin )^(-1)(x+(sin )^(-1)(x+(sin )^(-1)(x+...))). How do you find the amplitude and the period, of this FCS wave?

## Definition of ${\left(\sin\right)}^{- 1} X$: $Y = {\left(\sin\right)}^{- 1} X = k \pi + {\left(- 1\right)}^{k} {\sin}^{- 1} X , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots , Y \in \left[k \pi - \frac{\pi}{2} , k \pi + \frac{\pi}{2}\right]$

Jul 10, 2018

Axis of the wave: x + y = 0. Wave is bounded by the parallels $x + y = \pm 1$. Easily the amplitude is half of the width, $\frac{1}{\sqrt{2}}$ and the period is $2 \sqrt{2} \pi$.

#### Explanation:

Definition of piecewise-wholesome ${\left(\sin\right)}^{- 1} X$:

$Y = {\left(\sin\right)}^{- 1} X = k \pi + {\left(- 1\right)}^{k} {\sin}^{- 1} X ,$

$k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots , Y \in \left[k \pi - \frac{\pi}{2} , k \pi + \frac{\pi}{2}\right]$.

https://socratic.org/questions/on-what-interval-is-the-identity-sin-1-sin-x-x-valid639442

The FCS wave, with axis $x + y = 0$ and demarcation, marked by

$x + y = \pm 1$, is created by

$x = \sin y - y$. See uniform-scale graph. Observe the points of

inflexion aligned in x + y = 0 and crests and troughs aligned in

$x + y = \pm 1$.
graph{(x-sin y + y)(x+y)((x+y)^2-1)=0}

Graph of just one wave from $y = {\sin}^{- 1} \left(x + y\right)$ instead:

graph{(y-arcsin (x+y))(x+y)((x+y)^2-1)=0} .

The points of the meet with the axis are

$\pm \left(k \pi , - k \pi\right) , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

The amplitude = half-width between $x + y = \pm 1$ is $\frac{1}{\sqrt{2}}$.

Period = 2 ( distance between two consecutive axial points)

$= 2 \sqrt{2} \pi$.

Plots revealing a crust, a trough, period length and width.

graph{(x-sin y + y)(x-y)((x+y)^2-1)((x-3.14)^2+(y+3.14)^2-.01)((x+3.14)^2+(y-3.14)^2-.01)((x+0.57)^2+(y-1.57)^2 -.01)((x-0.57)^2+(y+1.57)^2 -.01)((x-0.5)^2 +(y-0.5)^2-.01)((x+0.5)^2 +(y+0.5)^2-.01)=0[-7 7 -3.5 3.5]}

Also see

https://socratic.org/questions/defining-the-wholesome-inverse-operator-sin-1-by-y-sin-1-x-k-pi-1-k-sin-1-x-y-in63919