how to get a nice "cosine looking" curve following the y=x direction?

Question

My aim is to get a "cosine looking" curve rotated 45° counterclockwise.

When I graph : g(x)= x + cos(x) , I get a curve that has lost the nice and regular wavering of the ordinary f(x)= cos(x) curve.

Adding a coefficient does not work, but rather aggravates the change of form.

Is there a possible equation that would produce the curve I am aiming at?

Thanks for your help.

Answer 1

Unfortunately, you can't with an explicit equation $y=f(x)$.

You need to resort to a parametric form such as

$$\begin{cases}x=t+\cos t,\\y=t-\cos t\end{cases},$$ obtained by rotation.

It is not possible to invert $x=t+\cos t$ analytically.

---

It is also possible to approximate this effect by assembling replicas and symmetries of the graph of a function like

$$y=\sqrt[n]{1-x^n}$$ computed in the interval $[0,1]$.

Answer 2

If you have the points of the form $\bigl(x,\cos(x)\bigr)$, you have the graph of the $\cos$ function. Now, multiply this by the matrix$$\begin{bmatrix}1&-1\\1&1\end{bmatrix}\left(=\sqrt2\begin{bmatrix}\cos\left(\frac\pi4\right)&-\sin\left(\frac\pi4\right)\\\sin\left(\frac\pi4\right)&\cos\left(\frac\pi4\right)\end{bmatrix}\right).$$In other words, consider the points of the form $\bigl(x-\cos(x),x+\cos(x)\bigr)$.

Answer 3

Wanting to consider the general case of roto-translation, writing in Wolfram Mathematica 12.0:

R[θ_] := {{Cos[θ], Sin[θ]}, {-Sin[θ], Cos[θ]}}
{xC, yC} = {2, 1};

xAxis = {xC, yC} + {t, 0}.R[θ];
yAxis = {xC, yC} + {0, t}.R[θ];
fun = {xC, yC} + {t, Cos[t]}.R[θ];

frames = Table[Magnify[ParametricPlot[{xAxis, yAxis, fun}, {t, -12, 12},
                       Epilog -> {Text[StringJoin["θ = ", ToString[TraditionalForm[θ]]],
                       {5, 9}], Black, PointSize[Large], Point[{xC, yC}]},
                       PlotRange -> {{-10, 10}, {-10, 10}},                                   
                       PlotStyle -> {Blue, Red, Green}], 2],
               {θ, 0, 2π, π/20}];

Export["image.gif", frames, "AnimationRepetitions" -> ∞, "DisplayDurations" -> 1];

we get:

Answer 4

As pointed out by user76284 in a comment, this can be done by writing an equation like $y-x=\cos(x+y)$. This form helps us to understand why we can't write down a closed-form equation for $y$ in terms of $x$, using only elementary functions. Such an equation would be a solution to a transcendental equation, which normally can't be expressed in this way. It also shows how we could probably write down a closed-form equation if we were willing to resort to special functions. For example, I would guess that you could write this function in terms of the Lambert W function or something similar.