Question

If $a\cos \theta -b\sin \theta =c,$ then find the value of $a\sin \theta +b\cos \theta .$

Answer 1

Given:

$a\cos \theta -b\sin \theta =c$

On squaring both the sides, we get

$a^2{\cos }^2\theta +b^2{\sin }^2\theta -2ab\cos \theta \sin \theta =c^2$ ..... (1)

Let $a\sin \theta +b\cos \theta =k$ (say)

On squaring both the sides, we get

$b^2{\cos }^2\theta +b^2{\sin }^2\theta +2ab\sin \theta \cos \theta =k^2$ ...... (2)

Add equations (1) and (2).

$a^2+b^2=c^2+k^2$

$k^2=a^2+b^2-c^2$

$k=\sqrt{a^2+b^2-c^2}$

Therefore,

$a\sin \theta +b\cos \theta =\sqrt{a^2+b^2-c^2}$