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-

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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