Question

From the top of aspire the angle of depression of the top and bottom of a tower of height h are $\theta$ and $\varphi$ respectively. Then height of the spire and its horizontal distance from the tower are respectively.

• A. $\frac{h\cos \theta \sin \varphi }{\sin (\varphi -\theta )}$ and $\frac{h\cos \theta \cos \varphi }{\sin (\varphi -\theta )}$
• B. $\frac{h\cos \theta \cos \varphi }{\sin (\theta +\varphi )}$, $\frac{h\tan \theta cos\varphi }{\sin (\theta +\varphi )}$
• C. $\frac{h\sin \theta sin\varphi }{\sin (\theta +\varphi )}$, $\frac{h\cos \theta \cos \varphi }{\sin (\theta +\varphi )}$
• D. None of these

Answer 1

Answer: (A)

$\frac{h\cos \theta \sin \varphi }{\sin (\varphi -\theta )}$ and $\frac{h\cos \theta \cos \varphi }{\sin (\varphi -\theta )}$

Given that $\angle ECD=\theta ,\angle ECA=\varphi ,AD=h$

From Geometry $\angle ECD=\theta =\angle FDC,\angle ECA=\varphi =\angle BAC$

$\tan \theta =\frac{BC-h}{AB}$ .............. $\left(1\right)$

$\tan \varphi =\frac{BC}{AB}\Rightarrow BC=AB\tan \varphi$ .......... $\left(2\right)$

Substitute equation $\left(2\right)$ in $\left(1\right)$

$AB\tan \theta =AB\tan \varphi -h$

Distance betweeen tower and sphire $AB=\frac{h}{\tan \varphi -\tan \theta }=\frac{h}{\frac{\sin \varphi }{\cos \varphi }-\frac{\sin \theta }{\cos \theta }}=\frac{h\cos \theta \cos \varphi }{\sin (\varphi -\theta )}$

Height of the spire $BC=AB\tan \varphi =\frac{h\cos \theta \cos \varphi }{\sin (\varphi -\theta )}\times \tan \varphi =\frac{h\cos \theta \sin \varphi }{\sin (\varphi -\theta )}$