Range of a function $|\sin x|+|\cos x|$ What is the range of function $Y=[|\sin x|+|\cos x|]$ where $[ ]$ denotes the greatest integer function.
And what is range of function $Y=|\sin x|+|\cos x|$
For the second one, 
I have tried squaring :
I got  $Y^{2}=1+|\sin 2x|$
and range will be between $1$ and $\sqrt{2}$
Please help me with this...!!! 

The most intuitive way to proceed might be to sketch the shape of the level sets
$$ |x|+|y| = 0 \\
|x|+|y| = 1  \\
|x|+|y| = 1.5\\
|x|+|y| = 2 $$
and so forth. Which of them have points in common with the unit circle?
