A ladder is Leaning against a vertical wall such that the top is nine meters above the ground and the base is three meters from the bottom of the wall. Find the measure of the angle between the ladder and the ground. Give your answer to two decimal places.
Let’s begin by sketching a diagram of this scenario. Remember a sketch does not need to be to scale, but it should be roughly in proportion so you can check the suitability of any answers you get.
The top of the ladder is nine meters above the ground and its base is three meters from the bottom of the wall. We can assume also that the angle between the ground and the wall is 90 degrees. Let’s call the angle we’re trying to find 𝜃. It’s the angle between the ladder and the ground.
So we have a right-angled triangle with two known lengths in which we’re trying to find the value of an angle. We need to use right angle trigonometry to do this. We can start by labelling the sides of the triangle. The hypotenuse is the longest side. It’s the side situated directly opposite the right angle. The opposite side is the side opposite to the given angle. It’s the one furthest away from 𝜃. Finally, the adjacent side is the other side. It’s located next to the angle 𝜃.
We can see that we know both the length of the opposite and the adjacent sides. That means we need to use the tan ratio. Tan 𝜃 is equal to opposite over adjacent. Since the length of the opposite side is nine meters and the adjacent is three, we can substitute these values into our formula, giving us tan 𝜃 is equal to nine over three. Nine divided by three is simply three. So our equation becomes tan 𝜃 is equal to three.
To solve this equation, we need to find the inverse tan of both sides. The inverse tan of tan 𝜃 is simply 𝜃. Our equation becomes 𝜃 is equal to the inverse tan of three. Substituting these values into our formula gives us 𝜃 is equal to 71.565.
The measure of the angle between the ladder and the ground is, therefore, 71.57 degrees correct to two decimal places.