TITLE: Skewed Trigonometric Function
QUESTION [8 upvotes]: What would be an expression for a periodic function (period $2\pi$) that essentially behaves just like a negative sine function, but it has the following quirk:
It's $0$s lie on the usual places (even integer multiples of $\frac \pi 2$), but it's maximum and minimum values (of $\pm 1$), instead of lying on odd integer multiples of $\frac\pi 2$, lie deviated by an angle $\alpha$ from the even integer multiples of $\frac \pi 2$. These are it's only maximums and minimums.
It's maximums and minimums can be graphically represented as follows:

In Orange we can see the function $-\sin (x)$, and the Red points represent the maximums and minimums of the function (the Gray lines just represent connections between the points, not the actual function).
Much appreciated.

REPLY [3 votes]: Considering the variables x, y, skew, t, α and k, and setting:

(x, y, skew, t, α) ∊ ℝ
k ∊ ℤ
f = (x, y) = (t + skew × sin t, - sin t), a parametric equation
α = skew + π/2

We get for skew = (0, -1, -π/2, 2) ⇒ α = (π/2, -1 + π/2, 0, 2 + π/2)
and 0 ≤ t ≤ 2π :

Note that ∫f = ∫(x, y) = (t + skew × sin t, (1/2) × (cos t) × (2 + skew × cos t)). For 0 ≤ t ≤ π the area above the curve is always 2, independent of the value of skew.
Note also that the function minima (-1) occurs at t = π/2 + 2kπ (as y = - sin t), for any value of skew.
As an example, for skew = 5π/32 and t = π/2 we get: