Sal solves a modeling problem where he finds the necessary angle to shoot at a vicious alien. Created by Sal Khan.
A tiny but horrible alien is standing at the top of the Eiffel Tower-- so this is where the tiny but horrible alien is-- which is 324 meters tall-- and they label that, the height of the Eiffel Tower-- and threatening to destroy the city of Paris. A Men In Black-- or a Men In Black agent. I was about to say maybe it should be a man in black. A Men In Black agent is standing at ground level, 54 meters across the Eiffel square. So 54 meters from, I guess you could say the center of the base of the Eiffel Tower, aiming his laser gun at the alien. So this is him aiming the laser gun. At what angle should the agent shoot his laser gun? Round your answer, if necessary, to two decimal places. So if we construct a right triangle here, and we can. So the height of this right triangle is 324 meters. This width right over here is 54 meters. It is a right triangle. What they're really asking us is what is this angle right over here. And they've given us two pieces of information. They gave us the side that is opposite the angle. And they've given us the side that is adjacent to the angle. So what trig function deals with opposite and adjacent? And to remind ourselves, we can write, like I always like to do, soh, cah, toa. And these are really by definition. So you just have to know this, and soh cah toa helps us. Sine is opposite over hypotenuse. Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. We can write that the tangent of theta is equal to the length of the opposite side-- 324 meters-- over the length of the adjacent side-- over 54 meters. Now you might say, well, OK, that's fine. What angle, when I take its tangent, gives me 324/54? Well, for this, it will probably be useful to use a calculator. And the way that we'd use a calculator is we would use the Inverse Tan Function. So we could rewrite this as we're going to take the inverse tangent-- and sometimes it's written as tangent with this negative 1 superscript. So the inverse tangent of tan of theta is going to be equal to the inverse tangent of 324/54. And just to be clear, what is this inverse tangent? This just literally says, this will return what is the angle that, when I take the tangent of it, gives me 324/54. This says, what is the angle that, when I take the tangent of it, gives me tangent of theta? So this right over here, this just simplifies to theta. Theta is the angle that when you get the tangent of it gets you tangent of theta. And so we get theta is equal to inverse tangent of 324/54. Once again, this inverse tangent thing you might find confusing. But all this is saying is, over here, we're saying tangent of some angle is 324/54. This is just saying my angle is whatever angle I need so that when I take the tangent of it, I get 324/54. It's how we will solve for theta. So let's get our calculator out. And let's say that we want our answer in degrees. Well, I'm just going to assume that they want our answers in degrees. So let me make sure my calculator is actually in degree mode. So I'll go to the 2nd mode right over here. And actually it's in radian mode right now. So let me make sure I'm in degree mode to get my answer in degrees. Now let me exit out of here. And let me just type in the inverse tangent-- so it's in this yellow color right here-- inverse tangent of 324 divided by 54 is going to be-- and they told us to round to two decimal places-- 80.54 degrees. So theta is equal to 80.54 degrees. That's the angle at which you should shoot the gun to help defeat this horrible alien.