First slide

Trigonometric equations

Question

Number of integer values of n so that $\sin x (\sin x + \cos x) = n$ has at least one solution

Moderate

A

0
B

1
C

2
D

3

Solution

$W e h a v e \sin^{2} x + \sin x \cos x = n$
$\Rightarrow \frac{1 - \cos 2 x}{2} + \frac{1}{2} \sin 2 x = n$
$\Rightarrow \sin 2 x - \cos 2 x = 2 n - 1$
$\Rightarrow - \sqrt{2} \leq 2 n - 1 \leq \sqrt{2} (∵ - \sqrt{2} \leq \sin 2 x - \cos 2 x \leq \sqrt{2})$
$\Rightarrow - \sqrt{2} + 1 \leq 2 n \leq \sqrt{2} + 1$
$\Rightarrow \frac{- \sqrt{2} + 1}{2} \leq n \leq \frac{\sqrt{2} + 1}{2} \Rightarrow n = 0, 1$

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