Question

If the length of the shadow of a tower is $\sqrt 3 $ times that of its height, then the angle of elevation of the sun is:
(A) ${15^ \circ }$
(B) ${30^ \circ }$
(C) ${45^ \circ }$
(D) ${60^ \circ }$

Answer 1

Hint: Start with assuming the height of the tower as some variable. Use the relation between the height of the tower and the length of its shadow given in the question to find the length of the shadow in terms of this variable. Finally apply the formula $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base }}}}$ to find the angle of elevation.

Complete step-by-step answer:

Consider the above diagram. Let AB is the tower and BC is its shadow. The angle of elevation of the sun is $\angle CAB = y$ as shown.
Further let the height of the tower is $x$.
$ \Rightarrow BC = x{\text{ }}.....{\text{(1)}}$
Then according to the question, the length of the shadow is $\sqrt 3 $ times the height of the tower. Thus:
$ \Rightarrow AB = \sqrt 3 x{\text{ }}.....{\text{(2)}}$
Now, for the triangle ABC, applying the formula $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base }}}}$ for $\angle CAB$, we’ll get:
$ \Rightarrow \tan y = \dfrac{{BC}}{{AB}}$
Putting the values of lengths of BC and AB from equation (1) and (2), we’ll get:
$ \Rightarrow \tan y = \dfrac{x}{{\sqrt 3 x}}$
Simplifying it further, we’ll get:
$ \Rightarrow \tan y = \dfrac{1}{{\sqrt 3 }}$
We know that $\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}$. So, on putting its value in the above equation, we’ll get:
$
   \Rightarrow \tan y = \tan {30^ \circ } \\
   \Rightarrow y = {30^ \circ } \\
$

Thus the angle of elevation of the sun is ${30^ \circ }$.

Note: The formula of $\tan \theta $ is used in a triangle whenever a relation between two non-hypotenuse sides of the triangle is required. Other trigonometric ratios can also be used if hypotenuse is to be determined or a relation between hypotenuse and any other side is to be compared.