A person in an office building looks out of a window from a height of six meters. They see a cat on the sidewalk 10 meters from the building. From the person, what is the angle of depression of the cat? Assume the sidewalk is horizontal.
Whenever there is a lot of detail in a question, it can be really useful to draw a diagram to represent the given information. Here we’re given the height of the person and the distance the cat is from this building. The question also tells us to assume the sidewalk is horizontal, which means that the angle between the sidewalk and the building must be 90 degrees.
The hypotenuse of this right-angled triangle represents the person’s line of sight. We’re interested in this because we’re being asked to find the angle of depression from the person. We also know that alternate angles are equal. Therefore, we can calculate the angle of depression by calculating the value of 𝜃 in this right-angled triangle.
Let’s start by labelling our sides. We can see that we know the value of both the opposite and the adjacent side. We are therefore going to need to use the tan ratio. Substituting our given lengths into this ratio gives tan 𝜃 equals six over 10. To solve this equation, we need to find the inverse of tan. 𝜃 is equal to inverse tan of six over 10. 𝜃 is therefore 30.96 degrees correct to two decimal places.
The angle of depression of the cat from the person is 30.96 degrees.