Question

The angle of elevation of a ladder leaning against a wall is ${60^ \circ }$ and the foot of the ladder is $9.5$m away from the wall, find the length of the ladder.

Answer 1

Hint: In this question we will try to figure out a right-angled triangle whose one of the angles is known to us. Then we will use the suitable trigonometric function to find the length of the ladder

Complete step-by-step answer:
Let the length of the ladder be L.
Given the angle of elevation of a ladder leaning against the wall is ${60^ \circ }$ and the distance between the foot of the ladder to the wall is $9.5$m.
That is according to the above figure we can say,
$BC = 9.5m$ -(1)
$AC = L$ -(2)
Now looking at the figure we can see a right-angled triangle ABC having a right angle at C and one of its angles is ${60^ \circ }$. So, now we can use any suitable trigonometric function for this triangle.
So,
$
  \cos \theta = \dfrac{{base}}{{hypo.}} \\
  \cos {60^ \circ } = \dfrac{{BC}}{{AC}} \\
 $
Now using (1) and (2) we get,
$\cos {60^ \circ } = \dfrac{{9.5}}{L}$
Using $cos{60^ \circ } = \dfrac{1}{2}$,
$
  \dfrac{1}{2} = \dfrac{{9.5}}{L} \\
  L = 9.5 \times 2 \\
  L = 19m \\
 $

So, the length of the ladder is 19m.

Note: In this question we have used trigonometric function $\cos \theta $ but here we do not need to memorize this, we only need to look at the information given and what we want to find. Like in this question we are given the base of the triangle and we need to find the hypotenuse and also, we are given the corresponding angle. So, according to this we can say $\cos \theta $ and $\sec \theta $. Here we used $\cos \theta $ but we can also use $\sec \theta $.