Math problem where graphing calculator may slow you down - part II.

[Bunuel66]

By hand and a bit of brain....
f(x) is periodic with a 2pi period.
from 0 to pi/4 f(x)=cos(x)
from pi/4 to 5pi/4 f(x)=sin(x)
from 5pi/4 to 2pi f(x)=cos(x)

Integrating is straightforward and gives 4/sqrt(2) over 0 to 2pi
going up to 9pi/4 gives 5/sqrt(2).
Then the value searched for is between 9pi/4 and 9pi/4+pi, in other words on the second interval of the definition of f(x).
This equivalent to search for X with: sum from pi/4 to X of sin(x) dx=4-5/sqrt(2).
Integrating analytically and solving lead to: X=acos(6/sqrt(2)-4)
The solution is then 9pi/4+X-pi/4~7.6089

Nothing more necessary than any standard scientific calculator...
BTW, I'm cheating slightly, I checked the value of the integral with my own python software ;-)
which uses a HP like algorithm....