A woman stands six feet from a vertical wall. A foot light on the ground three feet from where she is standing is switched on. If the woman is five feet tall and her shadow is 15 feet tall, what angle does the light make with the horizontal? Give your answer to two decimal places.
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.
We can assume that both the woman and the wall make an angle of 90 degrees with the horizontal. Notice that we have two similar triangles. One is an enlargement of the other. We’re interested in finding the angle the light makes with the horizontal. That’s 𝜃 in our diagram. We can use either triangle to calculate this value.
The small triangle is five foot tall and three foot wide. The larger triangle is 15 foot tall and its width can be calculated by adding three and six to get nine foot.
Let’s consider the smaller triangle. We have a right-angled triangle with two known lengths in which we’re trying to find an angle. We need to use right angle trigonometry to do this. Let’s label each side 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 the given angle. It’s the one furthest away from 𝜃. And 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. This means we need to use the tan ratio. tan 𝜃 is equal to opposite over adjacent. In our diagram, the opposite is five foot and the adjacent is three. That means then that tan 𝜃 is equal to five-thirds.
We solve this equation by finding the inverse tan of both sides. The inverse tan of tan 𝜃 is simply 𝜃, so 𝜃 is equal to the inverse tan of five-thirds. That’s 59.036. Correct to two decimal places, the angle that the light makes with the horizontal is 59.04 degrees.
Remember, since these two triangles are similar, we would have got the same answer had we substituted in the values from the larger triangle. That’s 15 and nine. Both 15 and nine can be divided by three, which once again gives us tan 𝜃 is equal to five-thirds. Solving this as we did earlier gives us 59.04 degrees.