A person observes a point on the ground from the top of a hill that is 1.56 kilometers high. The angle of depression is 29 degrees. Find the distance between the point and the observer giving the answer to the nearest meter.
Anytime we have a problem like this, it’s always helpful to draw an image. We have a person at the top of a hill looking down at a point on the ground. The person is 1.56 kilometers high. We have to be careful with the next piece of information. If we imagine that there’s a parallel line that runs from the place where the person is observing, the angle of depression is here, 29 degrees.
In order for us to do any kind of calculation, we can create a right triangle where the right angle is the perpendicular angle between the height and the distance across the ground. In order for us to identify angles, this angle would be alternate interior and would, therefore, be 29 degrees as well. And the two angles at the top are complementary. That means they must sum to 90 degrees, which makes this angle 61 degrees. Once we have all of this information, we can use our trig ratios to solve for the distance between the point and the observer.
We use the acronym SOH CAH TOA to remind us the sine ratio is opposite over the hypotenuse, the cosine ratio is the adjacent side over the hypotenuse, and the tangent ratio is the opposite over the adjacent. The distance between the point and the observer is this distance. It is opposite the right angle and is, therefore, the hypotenuse of this right triangle.
Now, we have another side that is 1.56 kilometers. But in order for us to label this side, we first need to choose which angle we’ll be using for the ratio. If we use the 29-degree angle, this is the opposite side. And if we use the 61-degree angle, this is the adjacent side. Let’s compare what the difference would be.
If we’re considering 61 degrees is the angle, we have the adjacent side length and we need the hypotenuse. Therefore, we would use the cosine relationship. We would say cos of 61 degrees equals 1.56 over 𝑑. If we use the 29-degree angle, we have the opposite side length. And we would say sin of 29 degrees equals 1.56 over 𝑑. Both of these will yield the same answer. And it will take us the same steps to solve both of them. But we’ll just solve one. We’ll solve sin of 29 degrees equals 1.56 over 𝑑.
To get the 𝑑 by itself, we’ll need to first get it out of the denominator. And we do that by multiplying both sides of the equation by 𝑑, which will give us 𝑑 times sin of 29 degrees equals 1.56. And from there, we divide both sides of the equation by sin of 29 degrees. We plug into our calculator 1.56 divided by the sin of 29 degrees. And we make sure that our calculator is set to degree mode and not to radians, which gives us 𝑑 equals 3.217757 continuing.
Now is where we need to pay close attention. This 1.56 was a measure of kilometers, which means that what we have now is a distance that’s measured in kilometers. Since we’re interested in having our final answer to the nearest meter, we know that one kilometer equals 1000 meters. And so, we multiply this 3.217757 continuing by 1000. And we get 3217.757 continuing. To round to the nearest meter, we’ll look to the right. There’s a seven in the tenths place. So, we round up to 3218 meters. The observer is 3218 meters from the point they’re observing.