Triple "Pythagorean identity" It's not hard to find multiple trigonometric functions of period $2\pi$ that added to self shifted by some constant offset result in a constant.
In classic pythagorean identity, you have 
$$F(x)+F\left(x+\frac{\pi}{2} \right) = 1 $$
where $F(x)=\sin^2 x$
Or you can use symmetry of the sine wave and create 
$F(x)+F(x+\pi ) = 0$
where $F(x) = \sin x$
Now what I'm looking for is a transformation of the sine function that while retaining the $2\pi$ period, gives you identity if repeated three times, with $\frac 2 3 \pi$ shift in each appearance:
$$F(x) + F\left(x + \frac 2 3 \pi \right) + F\left(x+\frac 4 3 \pi\right) = \mathrm{const}$$
Can you find such a function?
Reason and purpose:
I've been trying to develop a better RGB$<=>$HSV color space conversion - all the common ones use sawtooth style variant functions with variant equations, and I think using trigonometric functions could result in more smooth color passages, never mind much simpler algorithm.

Let $F(x)=\sin(ax+b)$ where a,b are indeterminate constants.
$F(x) + F\left(x + \frac 2 3 \pi \right) + F\left(x+\frac 4 3 \pi\right)  $
$=\sin(ax+b)+\sin(a(x+\frac{2\pi}{3})+b)+\sin(a(x+\frac{4\pi}{3})+b)$
$=\sin(ax+b)+\sin(a(x+\frac{4\pi}{3})+b)+\sin(a(x+\frac{2\pi}{3})+b)$
$=(1+2\cos\frac{2\pi a}{3})\sin(a(x+\frac{2\pi}{3})+b)$
This expression can not be constant for all $x$ unless  $1+2\cos\frac{2\pi a}{3}=0$
Or, $\cos\frac{2\pi a}{3}=-\frac{1}{2}=\cos\frac{2\pi}{3}$
$\implies \frac{2\pi a}{3}=2m\pi±\frac{2\pi}{3}$ where $m$ is an integer,
$\implies a=3m±1$
$F(x)=\sin(ax+b)$ where $a$ is an integer with $(a,3)=1$.
As $b$ is an indeterminate constant, we can have some other indeterminate constant $c=b-\frac{\pi}{2}$
$F(x)=\sin(ax+\frac{\pi}{2}+c)=\cos(ax+c)$
