Let a and b non negative real numbers If sin x+acos x=b , express ∣ asin x−cos x∣ in terms of a and b.


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Question Number 119295 by bobhans last updated on 23/Oct/20

$L e t a a n d b n o n n e g a t i v e r e a l n u m b e r s$ $I f \sin x + a \cos x = b, e x p r e s s$ $∣ a \sin x - \cos x ∣ i n t e r m s o f a a n d b .$
	Answered by bemath last updated on 23/Oct/20

	$c o n s i d e r r e l a t i o n s a^{2} + 1 = (\sin^{2} x + \cos^{2} x) (a^{2} + 1)$ $\Rightarrow a^{2} + 1 = a^{2} \sin^{2} x + \sin^{2} x + a^{2} \cos^{2} x + \cos^{2} x$ $= (a^{2} \sin^{2} x - 2 a \sin x \cos x + \cos^{2} x) +$ $(a^{2} \cos^{2} x + 2 a \cos x \sin x + \sin^{2} x)$ $= {(a \sin x - \cos x)}^{2} + {(a \cos x + \sin x)}^{2}$ $\Rightarrow {(a \sin x - \cos x)}^{2} = a^{2} + 1 - b^{2}$ $\Leftrightarrow a \sin x - \cos x = \pm \sqrt{a^{2} - b^{2} + 1}$ $\Leftrightarrow ∣ a \sin x - \cos x ∣ = \sqrt{a^{2} - b^{2} + 1}$
	Commented by bobhans last updated on 23/Oct/20

	$g r e a t!$
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