Question

If $\cos \alpha + \cos \beta =\frac{3}{2}$ and $\sin\,\alpha + \sin\,\beta =\frac{1}{2}$ and $\theta$ is the arithmetic mean of $\alpha$ and $\beta$ then $\sin\,2\theta + \cos\,2\theta$ is equal to :

• A. $\frac{3}{5}$
• B. $\frac{4}{5}$
• C. $\frac{7}{5}$
• D. $\frac{8}{5}$

Answer 1

Answer: (C)

$2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=\frac{3}{2}$ and $2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=\frac{1}{2}$
$\Rightarrow \tan \left(\frac{\alpha+\beta}{2}\right)=\frac{1}{3}$
$\Rightarrow \sin 2 \theta+\cos 2 \theta=\sin (\alpha+\beta)+\cos (\alpha+\beta)$
$=\frac{\frac{2}{3}}{1+\frac{1}{9}}+\frac{1-\frac{1}{9}}{1+\frac{1}{9}}=\frac{6}{10}+\frac{8}{10}$
$=\frac{7}{5}$