A five-meter ladder is leaning
against a vertical wall such that its base is two meters from the wall. Work out the angle between the
ladder and the floor, giving your answer to two decimal places.
We haven’t been given a diagram to
accompany this question. So the first step is going to be to
draw one. We have a ladder leaning against a
vertical wall. The triangle formed by the ladder,
the floor, and the wall is a right triangle. And in fact, that’s all we really
need to draw for our diagram. The ladder is five meters long, and
its base is two meters away from the wall. We are asked to work out the angle
between the ladder and the floor. So that’s this angle here, which
we’ll call 𝑥.
We have a right triangle in which
we know the lengths of two sides. And we want to calculate the
measure of an angle. So we can apply trigonometry. We begin by labeling the three
sides of the triangle in relation to the angle 𝑥. So we have the opposite, the
adjacent, and the hypotenuse. We then recall the acronym
SOHCAHTOA to help us decide which of the three trigonometric ratios to use in this
problem. The side lengths we know are the
adjacent and the hypotenuse. So we’re going to be using the
cosine ratio. This is defined for an angle 𝜃 as
the length of the adjacent divided by the length of the hypotenuse. Substituting two for the adjacent
and five for the hypotenuse, we have that cos of 𝑥 is equal to two-fifths.
To find the value of 𝑥, we need to
apply the inverse cosine function, giving 𝑥 is equal to the inverse cos of
two-fifths. Evaluating this on our calculators,
which must be in degree mode, gives 66.421. Finally, we round our answer to two
decimal places. And we found that the angle between
the ladder and the floor is 66.42 degrees.