Question

If from the top of a tower of $60$ metre high, the angles of depression of the top and floor of a house are $\alpha$ and $\beta$ respetivley and if the height of the house is $\frac{60\sin (\beta -\alpha )}{x}$, then $x=$

• A. $\sin \alpha \sin \beta$
• B. $\cos \alpha \cos \beta$
• C. $\sin \alpha \cos \beta$
• D. $\cos \alpha \sin \beta$

Answer 1

Answer: (D)

$\cos \alpha \sin \beta$

$\tan \alpha =\frac{60-h}{d}$

$\tan \beta =\frac{60}{d}$

$\frac{\tan \alpha }{\tan \beta }=\frac{60-h}{60}$

$\frac{\tan \alpha }{\tan \beta }=1-\frac{h}{60}$

$\frac{h}{60}=\frac{\tan \beta -\tan \alpha }{\tan \beta }$

$\frac{h}{60}=\frac{\sin \beta \cos \alpha -\sin \alpha \cos \beta }{\cos \alpha \sin \beta }$

$\frac{h}{60}=\frac{\sin (\beta -\alpha )}{\sin \beta \cos \alpha }$

$h=\frac{60\sin (\beta -\alpha )}{\cos \alpha \sin \beta }$

$\therefore x=\cos \alpha \sin \beta$