Law of cosines for star distance

Voiceover:Artemis seeks knowledge of

the width of Orion's belt,

which is a pattern of stars
in the Orion constellation.

She has previously
discovered the distances

from her house to
Alnitak, 736 lights years,

and to Mintaka, 915 light years,

which are the endpoints of Orion's belt.

She knows the angle between these

stars in the sky is three degrees.

What is the width of Orion's belt?

That is, what is the distance

between Alnitak and Mintaka?

And they want us to the
answer in light years.

So let's draw a little diagram

to make sure we understand
what's going on.

Actually, even before we do that,

I encourage you to pause

this and try this on your own.

Now let's make a diagram.

Alright, so let's say
that this is Artemis'

house right over here.

This is Artemis' house.

I'll say that's A for Artemis' house.

And then...

Alright, let me say H...

Let me say this is home.

This is home right over here.

And we have these 2 stars.

So she's looking out into the night sky

and she sees these stars,

Alnitak, which is 736 light years away,

and obviously I'm not going
to draw this to scale.

So this is Alnitak.

And Mintaka.

So let's say this is
Mintaka right over here.

Mintaka.

And we know a few things.

We know that this
distance between her home

and Alnitak is 736 light years.

So this distance right over here.

So that right over there.

Everything we'll do is in light years.

That's 736.

And the distance between

her house and Mintaka is 915 light years.

So it would take light 915 years

to get from her house to Mintaka,

or from Mintaka to her house.

So this is 915 light years.

And what we wanna do is figure out

the width of Orion's belt,

which is the distance
between Alnitak and Mintaka.

So we need to figure out this distance

right over here.

And the one thing that they did give us

is this angle.

They did give us that
angle right over there.

They said that the angle between

these stars in the sky is three degrees.

So this is three degrees right over there.

So how can we figure out the distance

between Alnitak and Mintaka?

Let's just say that this is equal to X.

This is equal to X.

How do we do that?

Well if we have two sides

and an angle between them,

we could use the law of cosines

to figure out the third side.

So the law of cosines,

so let's just apply it.

So the law of cosines tells us

that X squared is going to be equal

to the sum of the squares
of the other two sides.

So it's going to be equal to 736 squared,

plus 915 squared, minus two times 736,

times 915, times the cosine of this angle.

Times the cosine of three degrees.

So once again,

we're trying to find the length of

the side opposite the three degrees.

We know the other two sides,

so the law of cosines, it essentially...

Sorry, I just had to cough off camera

because I had some peanuts
and my throat was dry.

Where was I?

Oh, I was saying,

if we know the angle and
we know the two sides

on either side of the angle,

we can figure out the
length of the side opposite

by the law of cosines.

Where it essentially starts off not too

different than the Pythagorean theorem,

but then we give an adjustment

because this is not an
actual right triangle.

And the adjustment...

So we have the 736
squared, plus 915 squared,

minus two times the
product of these sides,

times the cosine of this angle.

Or another way we could
say, think about it is,

X, let me write that,

X is to equal to the square root of all

of this stuff.

So, I can just copy and paste that.

Copy and paste.

X is going to be equal to
the square root of that.

And so let's get our
calculator to calculate it.

And let me verify that I'm in degree mode.

Yes, I am indeed in degree mode.

And so let's exit that.

And so I wanna calculate
the square root of

736 squared, plus 915 squared,

minus two times 736, times 915,

times cosine of three degrees.

And we deserve a drum roll now.

X is 100, if we round...

Let's see, what did they want us to do?

Round your answer to
the nearest light years.

So to the nearest light year

is going to be 184 light years.

So X is approximately
equal to 184 light years.

So it would take light 184 years

to get from Mintaka to Alnitak.

And so hopefully this actually shows you

if you are going to do any astronomy,

the law of cosines, law of sines,

in fact all of trigonometry,

becomes quite, quite handy.