Question

If from the top of tower of $60$ metre height, the angles of depression of the top and floor of a house are $\alpha ,\beta $ respectively and if the height of the house is $\dfrac{{60\sin \left( {\beta - \alpha } \right)}}{x}$, then $x$ is
A. $\sin \alpha \sin \beta $
B. $\cos \alpha \cos \beta $
C. $\sin \alpha \cos \beta $
D. $\cos \alpha \sin \beta $

Answer 1

Hint: Choose different triangles and apply the formula of trigonometric ratios of sine, cosine and tangent. Simplify the trigonometric ratio and find the value of the height. Use the concept of triangles to get the answer.

Complete step-by-step answer:
Consider the diagram

In triangle $ABD$,
${{tan\beta }} = \dfrac{{{\text{DB}}}}{{{\text{AB}}}}$
${{tan\beta }} = \dfrac{{{\text{60}}}}{{\text{d}}}$
$d = 60\;{{cot\beta }}\;\;\;\;\;\left( {\therefore \;\dfrac{1}{{\tan \beta }} = \cot \beta } \right)......\left( 1 \right)$
Now in triangle $DEC$,
$\tan \alpha = \dfrac{{DE}}{{EC}}$
$\Rightarrow$$\tan \alpha = \dfrac{{DC}}{d}$
$\Rightarrow$$DC = d\tan \alpha ......\left( 2 \right)$
Here, $DC = 60 - h$ because $DC$ is calculated by subtracting $DB = h$.
Substitute the value of $DC = 60 - h$ and $d$ from equation (1).
Therefore,
$\Rightarrow$$60 - h = 60\cot \beta \tan \alpha ......\left( 3 \right)$
Now on simplifying equation (3).
$
\Rightarrow h = 60\left( {1 - \dfrac{{\cos \beta }}{{\sin \beta }} \cdot \dfrac{{\sin \alpha }}{{\cos \alpha }}} \right)\;\;\;\;\,\;\left( {\therefore \;\cot \beta = \dfrac{{\cos \beta }}{{\sin \beta }},\;\tan \alpha = \dfrac{{\sin \alpha }}{{\cos \alpha }}} \right) \\
    \\
 $
Take LCM and simplify it becomes,
$\Rightarrow$$h = \dfrac{{60\sin \left( {\beta - \alpha } \right)}}{{\cos \alpha \sin \beta }}$
$\Rightarrow$$h = \dfrac{{60\sin \left( {\beta - \alpha } \right)}}{x}\,\;\;\;\;\;\left( {\;given\;\cos \alpha \sin \beta = x} \right)$
Therefore,
$\dfrac{{60\sin \left( {\beta - \alpha } \right)}}{x} = \dfrac{{60\sin \left( {\beta - \alpha } \right)}}{{\cos \alpha \sin \beta }}$
$\Rightarrow$$x = \cos \alpha \sin \beta $

Hence, the option D is correct.

Note: Euclid and Archimedes presented the concept of Trigonometry. Trigonometry is a branch of mathematics that studies between side lengths and angles. Trigonometry helps to solve the problems of right-angled triangles.
Trigonometry ratios are the ratios between edges of the right-angle triangle. There are six trigonometric ratios sin, cos, tan, cosec, sec, cot. Sine function defined as the ratio of perpendicular to the hypotenuse. Cos function is defined as the ratio of base to the hypotenuse. Tan function is defined as the ratio of perpendicular to the base. The reciprocal of these functions defines cosec, sec and cot respectively.