Question

A ladder of length 6m leans against a vertical wall so that the base of the ladder is 2m from the wall. Calculate the angle between the ladder and the wall.

Answer 1

Hint:- Wall is always perpendicular to the ground. So, the triangle made by the ladder, wall and ground must be a right-angled triangle. And the sum of all angles of any triangle is equal to \[180^\circ \].

Complete step by step solution:
Let us first draw the figure for the given conditions in the question to make the problem easier.

So, as from the above figure we can see that AC will be the ladder and its length is 6m.
And BC will be the distance of the ladder from the foot of the wall and it is equal to 2m.
So, now we are asked to find the angle between the wall and the ladder.
And as we can see from the above figure that the angle between wall and the ladder is \[\angle BAC\].
And the triangle ABC is a right angled triangle. So, \[\angle ABC = 90^\circ \]
Now as we know that if the two sides of any triangle and one angle is given then we can find the other sides and angles of the right-angled triangle using trigonometric identities.
So, according to trigonometric identities in right angled triangle \[\sin \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}\]
So, in triangle ABC.
\[ \Rightarrow \sin \left( A \right) = \dfrac{{BC}}{{AC}} = \dfrac{2}{6} = \dfrac{1}{3}\]
So, taking sin inverse both sides of the above equation.
\[ \Rightarrow \angle A = \angle BAC = {\sin ^{ - 1}}\left( {\dfrac{1}{3}} \right)\]
Hence, the angle between ladder and wall will be equal to \[{\sin ^{ - 1}}\left( {\dfrac{1}{3}} \right)\].

Note:- Whenever we come up with this type of problem then first, we had to draw the figure using the given conditions in the question and after that if the triangle formed is right-angled triangle and its length of two sides and one angle is given then we can easily find the all other angles by simply using trigonometric formula like here we are given the length of the ladder (hypotenuse), distance between the wall and ladder (perpendicular) and we know that wall is perpendicular to ground. So, by using identity \[\sin \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}\] we can easily find the value of \[\theta \] for which perpendicular and hypotenuse is given. This will be the easiest and efficient way to find the solution of the problem.