Question

If $2cos\theta +sin\theta =1,\theta \ne \frac{\pi }{2}$, then the value of $7cos\theta +6sin\theta$ is equal to

• A. $\frac{1}{2}$
• B. $2$
• C. $\frac{11}{2}$
• D. $\frac{46}{5}$

Answer 1

The correct option is B $2$

$2cos\theta +sin\theta =1,\theta \ne \frac{\pi }{2}$
$\Rightarrow sin\theta =1-2cos\theta$
$\Rightarrow si{n}^2\theta =(1-2cos\theta {)}^2$
$\Rightarrow si{n}^2\theta =1+4co{s}^2\theta -4cos\theta$
$\Rightarrow 1-co{s}^2\theta =1+4co{s}^2\theta -4cos\theta$
$\Rightarrow 5co{s}^2\theta -4cos\theta =0$
$\Rightarrow cos\theta =\frac{4}{5}\Rightarrow sin\theta =1-\frac{8}{5}=\frac{-3}{5}$
$\therefore 7cos\theta +6sin\theta =\frac{28}{5}-\frac{18}{5}=\frac{10}{5}=2$