A rocket took off from a launch pad and traveled directly upwards. When the rocket was 15 miles up, one of the astronauts looked down at the mission control center, which is two miles away from the launch pad. What is the angle of depression from her to the mission control center? Round your answer to two decimal places.
So here we have our launch pad, and the astronaut in the rocket went 15 miles up, and then the astronaut looked down at the mission control center. So here our dashed line represents looking down at the mission control center, which is two miles away from the launch pad.
Now it’s asking, what is the angle of depression from her to the mission control center? So originally, if the astronaut would be looking directly out, so essentially she’s looking horizontal to the ground, and then it’s called an angle of depression because she has to lower her eyes in order to see the mission control center, and we wanna know what is the angle of which she had to look down, the angle of depression, from her where she was looking down to the mission control center.
So we have this right triangle because since they went directly upwards, it would be perpendicular with the ground; we have a right angle. So if we can find this angle 𝑥 in degrees, then that angle and the angle of depression that she had to look would add up to 90 degrees. So if we find 𝑥, we can subtract it from 90 and then we can find our missing angle of depression.
So focusing on our angle, we’re going to use either sine, cosine, or tangent. And to help remember, we use SOHCAHTOA: sine is equal to the opposite side divided by the hypotenuse side, cosine is equal to the adjacent side divided by the hypotenuse side, and tangent is equal to the opposite side divided by the adjacent side. So let’s see which sides we were given.
From our angle, the shortest side is the opposite side, two. The hypotenuse is always directly across from the right angle, the longest side, and we’re not given that one, so we won’t be using it. And then the 15 is right next to the angle, just like the hypotenuse, but it’s not the hypotenuse, so we call it the adjacent side.
So we have opposite and adjacent, so which of the three sine, cosine, or tangent uses opposite and adjacent? That would be tangent, so the tangent of our angle 𝑥 will be equal to the opposite side divided by the adjacent side, so two 15ths.
Now in order to solve for 𝑥, we need the inverse of tangent. So using our calculator, we use the inverse of tangent and then plug in two 15ths, and we get 7.5946 degrees.
Now it says to round to two decimal places for our final answer, so let’s go ahead and round this two decimal places. So we need to decide to whether keep the nine a nine or round it up to a 10. So we look at the number directly to the right of that; that’s four. And since four is less than five, we will keep the nine a nine.
So this angle that we found, 7.59 degrees, is on the inside of the triangle, and the angle of depression where the actual astronaut looked down is on the outside. And as we’ve stated before, those two angles should add to be 90 degrees, because traveling directly upward and then looking horizontally out would make a 90-degree angle.
So calling our missing angle of depression 𝑦, 7.59 degrees plus 𝑦 degrees equals 90 degrees, so let’s subtract 7.59 from both sides, and we get that our angle of depression is 82.41 degrees. So when the astronaut took off, went directly upward, and looked down at the control center, she had to look- make her eyes look down 82.41 degrees.