Solve for real numbers: 3 + sin(2x) = 4sin(x + (π/4))


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Question Number 156482 by MathSh last updated on 11/Oct/21

$Solve for real numbers :$ $3 + \sin (2 x) = 4 \sin (x + \frac{π}{4})$
	Answered by mr W last updated on 11/Oct/21

	$3 + \sin (2 x) = 2 \sqrt{2} (\sin x + \cos x)$ $9 + 6 \sin (2 x) + \sin^{2} (2 x) = 8 (1 + \sin (2 x))$ $\sin^{2} (2 x) - 2 \sin (2 x) + 1 = 0$ $\Rightarrow \sin (2 x) = 1 . . . (i)$ $3 + 1 = 4 \sin (x + \frac{π}{4})$ $\Rightarrow \sin (x + \frac{π}{4}) = 1 . . (i i)$ $x + \frac{π}{4} = 2 k π + \frac{π}{2}$ $\Rightarrow x = 2 k π + \frac{π}{4}$ $i t f u l f i l l s b o t h (i) a n d (i i) .$
	Commented by MathSh last updated on 11/Oct/21

	$Very nice solution dear S er, thank you$
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