Question

If $\sin \theta +2\cos \theta =1$ then find the value of $2\sin \theta -\cos \theta$ if $\sin \theta \ge 0$

Answer 1

Given - ${(sin\theta +2cos\theta )}^2=1^2$

$\Rightarrow {(\sin \theta +2\cos \theta )}^2+{(2\sin \theta -\cos \theta )}^2=1+{(2\sin \theta -\cos \theta )}^2$

$\Rightarrow$ Expand and solve

$\Rightarrow {5\sin }^2\theta +{5\cos }^2\theta =1+{(2\sin \theta -\cos \theta )}^2$

$\because {\sin }^{2}x+{\cos }^{2}x=1$

$\Rightarrow 5-1={(2\sin \theta -\cos \theta )}^2$

$\Rightarrow {(2\sin \theta -\cos \theta )}^2=4$

$\Rightarrow 2\sin \theta -\cos \theta =2\because \sin \theta \ge 0$