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Sign of Trigonometric Ratios in Different Quadrants

For which of ...

Question

For which of the following options, the equality $| sin x + cos x | = | sin x | + | cos x |$ holds TRUE?

x

can be an angle in

I

quadrant

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x

can be an angle in

III

quadrant

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x = n π, n \in Z

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x = (2 n + 1) \frac{π}{2}, n \in Z

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Solution

The correct option is D $x = (2 n + 1) \frac{π}{2}, n \in Z$
Since, $| a + b | = | a | + | b | \Rightarrow a b \geq 0$
So, $| sin x + cos x | = | sin x | + | cos x |$
$\Rightarrow sin x cos x \geq 0$

Possible cases are :
$(i) sin x$ and $cos x$ are of same sign.
i.e., $x \in I$ quadrant or $x \in III$ quadrant.

$(i i) sin x = 0 \Rightarrow x = n π, n \in Z$

$(i i i) cos x = 0 \Rightarrow x = (2 n + 1) \frac{π}{2}, n \in Z$

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