First slide

Trigonometric equations

Question

If $f : R \to S$ is given by $f (x) = \sin x - \sqrt{3} \cos x + 1$ is onto, then interval of S is

Moderate

A

$[0, 1]$
B

$[- 1, 1]$
C

$[0, 3]$
D

$[- 1, 3]$

Solution

$W e h a v e - \sqrt{a^{2} + b^{2}} \leq a \sin x + b \cos x \leq \sqrt{a^{2} + b^{2}}$
So $- \sqrt{1 + 3} \leq \sin x - \sqrt{3} \cos x \leq \sqrt{1 + 3}$
$\Rightarrow - 2 \leq \sin x - \sqrt{3} \cos x \leq 2$
$\Rightarrow - 2 + 1 \leq \sin x - \sqrt{3} \cos x + 1 \leq 2 + 1$
$\Rightarrow - 1 \leq f (x) \leq 3$
$\Rightarrow S = [- 1, 3]$

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