Question

If $2\cos \theta +\sin \theta =1(\theta \ne \frac{\pi }{2})$ then $7\cos \theta +6\sin \theta$ is equal to?

Answer 1

$2cos\Theta =1-sin\Theta$

$onsquaringboththesides$

$\Rightarrow 4co{s}^2\Theta =si{n}^2\Theta -2sin\theta +1$

$\Rightarrow 4(1-si{n}^2\Theta )=si{n}^2\Theta -2sin\Theta +1$

$\Rightarrow 5si{n}^2\Theta -2sin\Theta -3=0$

$\Rightarrow sin\Theta =1;sin\Theta =-\frac{3}{5}$

$Asperthequestion\theta \ne \frac{\pi }{2}$

$henceheresin\theta =-\frac{3}{5}sothecos\Theta =\frac{4}{5}$

$requiredanswer=7\left(\frac{4}{5}\right)+6(-\frac{3}{5})=2$