A 90-foot-tall building has a shadow that is two feet long. What is the angle of elevation of the Sun?
Let’s begin by sketching a diagram of this scenario. Remember, a sketch doesn’t need to be to scale, but it should be roughly in proportion so we can check the suitability of any answers we get.
The building has a height of 90 feet and its shadow is two feet long. We can assume that the angle between the building and its shadow is 90 degrees. We’re looking to find the angle of elevation of the Sun. In this case, that’s the angle between the horizontal and the line made between the end of the shadow and the top of the building. It’s this angle 𝜃.
So we have a right-angled triangle with two known lengths, in which we’re trying to find an angle 𝜃. We need to use right angle trigonometry to do this.
We’re gonna 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 is the side opposite the given angle. It’s the one furthest away from the angle 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 side. This means we need to use the tan ratio. Tan 𝜃 is equal to opposite over adjacent. Substituting the values from our triangle into the formula gives us tan of 𝜃 is equal to 90 divided by two, which is 45.
To solve this equation, we’ll find the inverse tan of both sides. The inverse tan of tan 𝜃 is simply 𝜃, so 𝜃 is equal to the inverse tan of 45. Inverse tan of 45 is 88.7269. Correct to two decimal places, the angle of elevation of the Sun is 88.73 degrees.