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 will begin by sketching a diagram to model the situation. We have a five-meter-long ladder leaning against a vertical wall. And we are told that the base of the ladder is two meters from the wall. Since the wall is vertical, these lengths create a right triangle. And we have been asked to work out the angle between the ladder and the floor. We will call this angle 𝜃. Our first step is to label the three sides of the right triangle as shown. The hypotenuse is opposite the right angle. The side opposite angle 𝜃 is known as the opposite, and the side next to angle 𝜃 and the right angle is the adjacent.
We are now in a position to use our knowledge of right angle trigonometry and the trigonometric ratios to calculate the measure of angle 𝜃. We can recall these ratios using the acronym SOH CAH TOA. In this question, we know the length of the adjacent and hypotenuse, so we will use the cosine ratio. This states that cos 𝜃 is equal to the adjacent over the hypotenuse. Substituting in the lengths from our diagram, we have cos 𝜃 is equal to two-fifths.
We can then take the inverse cosine of both sides such that 𝜃 is equal to the inverse cos of two-fifths. Ensuring that our calculator is in degree mode, we type this in, giving us 𝜃 is equal to 66.4218 and so on. We are asked to round this answer to two decimal places. Since the third digit after the decimal point is a one, we round down, giving us 66.42.
The angle between the ladder and the floor correct to two decimal places is 66.42 degrees.