A rocket is launched vertically upward. A woman, standing four miles from the launchpad, watches its flight. What is the angle of elevation of the rocket from the women when its altitude is 11 miles?
Let’s begin by sketching a diagram of this scenario. Remember, a sketch does not need to be to scale. But it should be roughly in proportion so we can check the suitability of any answers we get.
Let’s call the woman 𝐴 and the launchpad 𝐵. The distance between 𝐴 and 𝐵 is four miles. The rocket launches vertically upwards, which means that the angle between the trajectory of the rocket and the ground is 90 degrees. We’re interested in the moment that the rocket — let’s call that 𝐶 — is at an altitude of 11 miles. So the length 𝐵𝐶 is 11 miles.
We are looking to find the angle of elevation of the rocket from the woman. It’s the angle between the horizontal and the line from the rocket to the woman’s eye. That’s angle 𝐶𝐴𝐵, which we’ve called 𝜃. So we have a right-angled triangle with two known lengths, in which we’re trying to find a missing angle.
We’ll need to use right-angle trigonometry to do this. We can start by labelling 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 the angle 𝜃. Finally, the adjacent side is the other side. It’s located next to the angle 𝜃.
We know both lengths of the opposite and the adjacent sides. This means we need to use the tangent ratio. Tan of 𝜃 is equal to opposite over adjacent. The length of the opposite side of our triangle is 11 miles, and the length of the adjacent side is four. So our equation becomes tan 𝜃 is equal to 11 over four.
To solve this equation for 𝜃, we want to find the inverse tan of both sides. The inverse tan of tan of 𝜃 is simply 𝜃. So our equation becomes 𝜃 is equal to the inverse tan of 11 over four. If we type inverse tan of 11 over four into our calculator, we get 70.016, and so on. Correct to two decimal places, the angle of elevation of the rocket from the woman is 70.02 degrees.