Given that:
2*f(sin x) + f(cos x) = x
Find f'(x)
Such a function cannot exist for
all values of x.
Sine and cosine are periodic. Increasing the lhs by 2pi will leave it unchanged.
We want lhs to increase linearly. I know arcsin(sin(x) and arcsin(cos(x)) are zigzags of linear segments.
A little playing around gives f(x)=arcsin(x)pi/6
which is equal to x from 0 to pi/2
The derivative of arcsin(x)pi/6 is 1/sqrt(1x^2)
which on its own is only defined from 1 to 1.
https://www.desmos.com/calculator/ukmoaqzpuv
There are more solutions that can be found by messing with the slopes and shifting up or down. Here's one using arccosine and defined from pi/2 to pi. So now the derivative is 1/(3sqrt(1x^2))
It looks like by playing with the a parameter and arcsin/arccos there may be 8 solutions in all.

Posted by Jer
on 20240619 09:48:36 