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Domain

Solve |sin ...

Question

Solve $| sin x + cos x | = | sin x | + | cos x |, x \in [0, 2 π]$ .

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Solution

Given $| s i n x + c o s x | = | s i n x | + | c o s x |$
For $0 \leq x \leq π / 2$
$s i n x + c o s x = s i n x + c o s x$
$\Rightarrow 0 ⩽ x ⩽ π / 2$
For $π / 2 ⩽ x ⩽ π, s i n x > 0 c o s x < 0$
LHS $s i n x - c o s x > 0 x < 3 π / 4$
$s i n x - c o s x < 0 x > 3 π / 2$
RHS : $s i n x + c o s x$
L.H.S $\neq$ RHS
For $π ⩽ x < 3 π / 2$
LHS = $| - s i n x - c o s x |$
RHS = $| - s i n x | + | - c o s x |$
LHS = RHS $\Rightarrow π ⩽ x ⩽ 3 π / 2$
For $3 π / 2 < x < 2 π$
LHS : $| - s i n x + c o s x | = | c o s x - s i n x |$
RHS : $| s i n x | + | c o s x |$
LHS $\neq$ RHS
$\Rightarrow 0 ⩽ x ⩽ π / 2$ and $π ⩽ x ⩽ 3 π / 2$

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