Question

Solve |`sinx + cosx`| = |`sinx`| + |`cosx`|, x`in` [0, 2`pi`] ?

Answer 1

`x in [0,pi/2] cup [pi,(3pi)/2]`

Explanation:

For real numbers `x` and `y`, the relation

`|x+y| = |x|+|y|`

works as long as both `x` and `y` are both positive, or both negative.

This means that for

`|sin x+cos x| = |sin x|+|cos x|`

to hold :

• either both `sin x` and `cos x` are positive - hence `x` is in the first quadrant.
• or, both are negative - hence `x` is in the third quadrant.