Question

The angle of elevation of the top of a car from height at a distance of a & b meters. From the base of the tower and in the same straight line with it are complimentary. Find the height of tower ?

Answer 1

$\alpha +\beta =90$

$\alpha =90-\beta$

$\sin \alpha =\cos \beta -\left(i\right)$ and $\cos \alpha =\sin \beta -\left(ii\right)$

$\tan \alpha =\frac{a}{h}$ and $\tan \beta =\frac{b}{h}$

$\frac{i}{\left(ii\right)}\Rightarrow \tan \alpha =\frac{1}{\tan \beta }$

$\tan \alpha \tan \beta =1$

$\sin (\alpha +\beta )=\sin \left(90\right)=1$

$\sin \alpha \cos \beta +\cos \alpha \sin \beta =1$

Divide by $\cos \alpha \cos \beta$

$\tan \alpha +\tan \beta =1=\frac{a}{h}+\frac{b}{h}=1h=a+b$