Question

Find the angle of elevation of the when the length of the shadow of a vertical pole is equal to its height.

Answer 1

Hint: Assume that the height of the tower from the ground is ‘h’. First, draw a rough diagram of the given conditions and then use the formula $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$ in the different right angle triangles and substitute the given values to get the height.

Complete step-by-step answer:


Let us start with the question by drawing a representative diagram of the situation given in the question.

According to the above figure:

We have assumed the height of the tower from the ground as ‘h’. Therefore, AB = h. Also, assume that the distance BD is ‘x’ meters.

Now, in right angle triangle ADB,

We let $\angle ADB=\theta $ .

We know that, $\tan \theta =\dfrac{\text{perpendicular}}{\text{base}}$. And it is given that the height of the tower is equal to the length of the shadow. So, we can say that AB=BD.

$\therefore x=h$

$\begin{align}

  & \therefore \tan \theta =\dfrac{AB}{BD} \\

 & \Rightarrow \tan \theta =\dfrac{h}{x} \\

 & \Rightarrow \tan \theta =1 \\

\end{align}$

Now we know that the tangent of $45{}^\circ $ is 1. Therefore, we can conclude that $\theta $ is equal to $45{}^\circ $ .

So, the angle of elevation, i.e., the elevation of the sun is $45{}^\circ $.


Note: Do not use any other trigonometric function like sine or cosine of the given angle because the information which is provided to us is related to the base of the triangle and the height. So, the length of the hypotenuse is of no use. Therefore, the formula of the tangent of the angle is used. We can use sine or cosine of the given angles but then the process of finding the height will be lengthy.