Question

If the length and height of the shadow of a man are the same, then find the angle of elevation of the sun.

Answer 1

Hint: Here, we should know that shadow is always formed on ground i.e. base. So, we will get a figure like this.
 

Complete step-by-step solution -
After this, we have to assume height and length as some variable x. And then using trigonometric rule to find angle given as $\tan \theta =\dfrac{\text{Opposite}}{\text{adjacent}}$ and further substituting the values, we will get value of $\theta $ in degrees.

In the question we are given that height and length of shadow of man is same so, we can draw figure as

Now, as we can see that OP is height and PQ is length, so on joining OQ we get a right angled triangle. We will consider here that height and length to be equal to x. So, we get as

Now, we have to find the angle of elevation $\theta $ which is shown in the figure. So, we can use here the trigonometric formula of finding angle using the tan function given as $\tan \theta =\dfrac{\text{Opposite}}{\text{adjacent}}$ .
So, by using this we get
$\tan \theta =\dfrac{OP}{PQ}=\dfrac{x}{x}=1$
Now, taking tan inverse on both the sides we get
$\theta ={{\tan }^{-1}}\left( 1 \right)$
We know that $\tan \left( 45{}^\circ \right)=1$ , so here the value of $\theta $ will be equal to $45{}^\circ $ .
Thus, $\theta =45{}^\circ $ .
Hence, angle of elevation is $45{}^\circ $ .

Note: Another method for solving this is by using Pythagoras theorem and finding the third side i.e. $O{{P}^{2}}+P{{Q}^{2}}=O{{Q}^{2}}$ . After putting the values and solving the equation, we get $OQ=\sqrt{2}x$ . So, the figure will be like this.

Now, we can use any trigonometric function like $\sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}}$ or $\cos \theta =\dfrac{\text{Adjacent}}{\text{Hypotenuse}}$ . On substituting the value and taking inverse, we will get the same answer that is $45{}^\circ $ .