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### Course: Get ready for AP® Calculus>Unit 4

Lesson 6: Modeling with right triangles

# Right triangle word problem

Sal solves a modeling problem where he finds the necessary angle to shoot at a vicious alien. Created by Sal Khan.

## Want to join the conversation?

• Could someone explain again what is the difference between tan and tan-1? Why did Sal use tan-1 instead of tan?
(27 votes)
• hmm, it's very simple idea from functions. Function takes number from domain(the set of numbers for which you want the function to work on) , works on it , and gives you number in the codomain or range(both essentially mean the numbers you get from function). Inverse function takes numbers from codomain or range of the function, works on it, and gives you numbers in domain of the function.

For our example, tan is function of our angle. it takes angles, which is domain, and gives you their tangents(opp/adj ratio if they are angle in right triangle), which is the range or codomain. Tan^-1, also known as arctan, takes the tangents,which is codomain of tan, and gives you the angles, which is domain of tan.
Hope that helps :D
(63 votes)
• Hey why don't we just write tan^-1(6) instead writing tan^-1(324/54) . Would it be the same answer?
(11 votes)
• Yes, you are correct. Most teachers would expect you to simplify the fraction.
(14 votes)
• Isn't Cot is the inverse of Tan?
(5 votes)
• No. The cotangent is the reciprocal, not the inverse of tangent. The inverse tangent is the arctangent.
NOTE: There is some unfortunate difficulty in notation. If the symbol ⁻¹ comes immediately after the name or symbol of a function, then it does NOT mean the reciprocal: it means the inverse function. But if ⁻¹ comes AFTER some constant, variable or other expression, then it means the reciprocal.

So, again, tan⁻¹ x is NOT equal to 1/tan x and so it is NOT the cotangent.
(20 votes)
• What is the difference between a right and a left triangle?
(0 votes)
• Right triangles have a right angle. There are no left triangles.
(11 votes)
• What angle would he have to aim IF the laser beam was affected by gravity?
(4 votes)
• Laser beams are affected by gravity. But so minutely that the effect wouldn't appear in the second decimal place of the angle. So the answer would be the same.
(17 votes)
• This question is mostly relying on calculator. Can you teach us how to do it using your brain, without paper, pencil, or calculator?
(7 votes)
• if there is a angle given and a side, how do I solve
(10 votes)
• Sorry for the late reply, but I think to solve these types of questions you would need to apply things like the Sine rule, and the Cosine rule.
(1 vote)
• Technically, isn't the agent going to shoot the tip of the eiffel tower if he aims at 80.54 degrees? Yes, the alien is tiny but I think his height should be accounted for. Just a side thought.
(5 votes)
• Or maybe we are try to blow the top of the tower off so that the alien falls. Of course, if we think to hard about this question, we will end up realizing things like the fact that the agent didn't need to know the angle to shoot, he could have just pointed the gun straight at the alien.
(6 votes)
• (ANSWERED) Instead of using theta, could Sal use the right angle in the triangle for the equation?
(5 votes)
• No. When using the trig ratios in right triangles, theta refers to an angle less than 90 degrees. If you used the hypotenuse, there would be ambiguity as to which side was adjacent, and because tan(90 deg) = undefined, and cos(90 deg) = 0. Both of those answers don't make sense in a solving a right triangle point of view.
(7 votes)
• does anyone know how to solve a word problem with overlapping triangles?. Its basically like the one he showed but lets say there is another agent, the height of depression of that agent is 60 degrees. So, the question is, what is the distance between the two agents?
(5 votes)
• The height of depression does not make sense, so I have to make assumptions. I assume you mean that a second agent is standing on the ground further from the tower where the angle of elevation is 60 degrees. So we know the original agent is 54 meters away. We want to know how many meters the second agent is away, use tan(60)=324/x, if x is on bottom, switch places to get x=324/tan(60)=187m. So distance between would just be subtracting these two distances.
(3 votes)

## Video transcript

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.