First slide

Trigonometric equations

Question

$I f 0 \leq x \leq 2 π a n d |\cos x| \leq \sin x, t h e n$

Moderate

A

$x \in [0, \frac{π}{4}]$
B

$x \in [\frac{π}{4}, 2 π]$
C

$[\frac{π}{4}, \frac{3 π}{4}]$
D

$[0, π]$

Solution

$W e h a v e |\cos x| \leq \sin x \Rightarrow \sin x \geq 0 (∵ |\cos x| \geq 0)$
$\Rightarrow x \notin [π, 2 π]$
$I f x \in (0, \frac{π}{2}], t h e n |\cos x| \leq \sin x \Rightarrow \cos x \leq \sin x \Rightarrow x \in [\frac{π}{4}, \frac{π}{2}]$
$I f x \in (\frac{π}{2}, π), t h e n |\cos x| \leq \sin x \Rightarrow - \cos x \leq \sin x (∵ \cos x i s n e g a t i v e) \Rightarrow \tan x \leq - 1 \Rightarrow x \in [\frac{π}{2}, \frac{3 π}{4}] ∴ x \in [\frac{π}{4}, \frac{3 π}{4}]$

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