(C) Let $h$ be the height of the tower $OP$,where $O$ is the base of the tower.In $\triangle OAP$,$\tan \alpha = \frac{h}{OA} \Rightarrow OA = h \cot

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$\frac{d}{\sqrt{\cot^2 \alpha + \cot^2 \beta}}$

(C) Let $h$ be the height of the tower $OP$,where $O$ is the base of the tower.In $\triangle OAP$,$\tan \alpha = \frac{h}{OA} \Rightarrow OA = h \cot \alpha$.In $\triangle OBP$,$\tan \beta = \frac{h}{OB} \Rightarrow OB = h \cot \beta$.Since $OA$ is south and $OB$ is east,$\triangle OAB$ is a right-angled triangle at $O$.By Pythagoras theorem in $\triangle OAB$,$OA^2 + OB^2 = AB^2$.$(h \cot \alpha)^2 + (h \cot \beta)^2 = d^2$.$h^2 (\cot^2 \alpha + \cot^2 \beta) = d^2$.$h^2 = \frac{d^2}{\cot^2 \alpha + \cot^2 \beta}$.$h = \frac{d}{\sqrt{\cot^2 \alpha + \cot^2 \beta}}$.

Class 11 Mathematics Trigonometrical Equations Height and Distance

Medium

The angle of elevation of the top of a tower from a point $A$ due south of the tower is $\alpha$ and from a point $B$ due east of the tower is $\beta$. If $AB = d$,then the height of the tower is

A
$\frac{d}{\sqrt{\tan^2 \alpha - \tan^2 \beta}}$
B
$\frac{d}{\sqrt{\tan^2 \alpha + \tan^2 \beta}}$
C
$\frac{d}{\sqrt{\cot^2 \alpha + \cot^2 \beta}}$
D
$\frac{d}{\sqrt{\cot^2 \alpha - \cot^2 \beta}}$

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Height and Distance

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The angle of elevation of the top of a tower from a point $A$ due south of the tower is $\alpha$ and from a point $B$ due east of the tower is $\beta$. If $AB = d$,then the height of the tower is

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The angle of elevation of the top of a tower from a point $A$ due north of it is $\alpha$ and from a point $B$ at a distance of $9$ units due west of $A$ is $\cos^{-1}\left(\frac{3}{\sqrt{13}}\right)$. If the distance of the point $B$ from the tower is $15$ units,then $\cot \alpha$ is equal to.

$A$ tower,of $x$ metres high,has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant $y$ metres from the foot of the tower. Then,the length of the flagstaff (in metres) is:

If the angles of elevation of the top of a tower from three collinear points $A, B$ and $C$ on a line leading to the foot of the tower are $30^o, 45^o$ and $60^o$ respectively,then the ratio $AB : BC$ is:

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