Question

If the height of a tower and the length of its shadow is equal, then the value of the angle of elevation of the sun is-
(a) \[{{30}^{\circ }}\]
(b) \[{{45}^{\circ }}\]
(c) \[{{60}^{\circ }}\]
(d) None of these

Answer 1

Hint: We start solving the problem by drawing the figure representing the given information. We then recall the definition of the tangent of an angle in a right-angled triangle. We then take the tangent of the angle of elevation in the figure we just drew. We then make the necessary calculations to get the required value of the angle.

Complete step-by-step solution
According to the problem, we are given that the height of a tower is equal to the length of its shadow. We need to find the angle of the elevation of the sun.
Let us assume AB be the height of the tower and BC be the length of the shadow and the angle of elevation be $\theta $ and draw the figure representing the given information.

According to the problem, we are given $AB=BC$ ---(1).
We know that the tangent of an angle in a triangle is defined as the ratio of the opposite side to the adjacent side for the angle.
From triangle ABC, we have $\tan \theta =\dfrac{AB}{BC}$.
From equation (1), we get $\tan \theta =\dfrac{AB}{AB}$.
$\Rightarrow \tan \theta =1$.
$\Rightarrow \theta ={{\tan }^{-1}}\left( 1 \right)$.
$\Rightarrow \theta ={{45}^{\circ }}$.
So, we have found the value of the angle of elevation of the sun as \[{{45}^{\circ }}\].
The correct option for the given problem is (b).

Note: We can see that the given triangle is a right-angles isosceles triangle which makes the angle of depression is equal to the angle of elevation. Whenever we get this type of problem, we should first try to draw the figure representing the given information. We should know that the angle of elevation will be an acute angle. Similarly, we can expect problems to find the length of the shadow if the angle of elevation of the sun is changed to \[{{30}^{\circ }}\].