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## Trigonometry

### Unit 1: Lesson 6

Modeling with right triangles# Right triangle word problem

CCSS.Math:

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

## 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.