Question

If $2\cos \theta +\sin \theta =1(0\ne \pi /2)$, then find the value of $7\cos \theta +6\sin \theta$

Answer 1

$2cos\theta +sin\theta =1$

$\Rightarrow 2cos\theta =1-sin\theta$

$\Rightarrow 4co{s}^2\theta =(1-sin\theta {)}^2$

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

$\Rightarrow 4(1-sin\theta )(1+sin\theta )-(1-sin\theta {)}^2=0$

$\Rightarrow (1-sin\theta )(4+4sin\theta -1-sin\theta )=0$

$\Rightarrow (1-sin\theta )(3+3sin\theta )=0$

$\Rightarrow [sin\theta =1][sin\theta =-1]$

Sinece$\theta \ne 72$

$\therefore$ $sin\theta =-1$

$\therefore$ $cos\theta =0$

$7cos\theta +6sin\theta$

=7(0)+6(-1)

=-6