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Voiceover:Artemis seeks knowledge of

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the width of Orion's belt,

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which is a pattern of stars[br]in the Orion constellation.

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She has previously[br]discovered the distances

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from her house to[br]Alnitak, 736 lights years,

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and to Mintaka, 915 light years,

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which are the endpoints of Orion's belt.

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She knows the angle between these

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stars in the sky is three degrees.

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What is the width of Orion's belt?

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That is, what is the distance

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between Alnitak and Mintaka?

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And they want us to the[br]answer in light years.

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So let's draw a little diagram

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to make sure we understand[br]what's going on.

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Actually, even before we do that,

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I encourage you to pause

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this and try this on your own.

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Now let's make a diagram.

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Alright, so let's say[br]that this is Artemis'

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house right over here.

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This is Artemis' house.

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

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And then...

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Alright, let me say H...

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Let me say this is home.

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This is home right over here.

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And we have these 2 stars.

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So she's looking out into the night sky

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and she sees these stars,

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Alnitak, which is 736 light years away,

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

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So this is Alnitak.

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And Mintaka.

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

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Mintaka.

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And we know a few things.

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We know that this[br]distance between her home

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and Alnitak is 736 light years.

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So this distance right over here.

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So that right over there.

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Everything we'll do is in light years.

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That's 736.

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And the distance between

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her house and Mintaka is 915 light years.

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So it would take light 915 years

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to get from her house to Mintaka,

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or from Mintaka to her house.

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So this is 915 light years.

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And what we wanna do is figure out

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the width of Orion's belt,

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which is the distance[br]between Alnitak and Mintaka.

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So we need to figure out this distance

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right over here.

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And the one thing that they did give us

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is this angle.

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They did give us that[br]angle right over there.

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They said that the angle between

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these stars in the sky is three degrees.

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So this is three degrees right over there.

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So how can we figure out the distance

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between Alnitak and Mintaka?

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

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This is equal to X.

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How do we do that?

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Well if we have two sides

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and an angle between them,

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we could use the law of cosines

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to figure out the third side.

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So the law of cosines,

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so let's just apply it.

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So the law of cosines tells us

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that X squared is going to be equal

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to the sum of the squares[br]of the other two sides.

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

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plus 915 squared, minus two times 736,

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times 915, times the cosine of this angle.

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Times the cosine of three degrees.

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So once again,

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we're trying to find the length of

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the side opposite the three degrees.

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We know the other two sides,

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so the law of cosines, it essentially...

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Sorry, I just had to cough off camera

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because I had some peanuts[br]and my throat was dry.

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Where was I?

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Oh, I was saying,

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if we know the angle and[br]we know the two sides

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on either side of the angle,

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we can figure out the[br]length of the side opposite

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by the law of cosines.

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Where it essentially starts off not too

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different than the Pythagorean theorem,

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but then we give an adjustment

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because this is not an[br]actual right triangle.

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And the adjustment...

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So we have the 736[br]squared, plus 915 squared,

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minus two times the[br]product of these sides,

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times the cosine of this angle.

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Or another way we could[br]say, think about it is,

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X, let me write that,

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X is to equal to the square root of all

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of this stuff.

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So, I can just copy and paste that.

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Copy and paste.

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

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

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

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Yes, I am indeed in degree mode.

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And so let's exit that.

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And so I wanna calculate[br]the square root of

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736 squared, plus 915 squared,

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minus two times 736, times 915,

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times cosine of three degrees.

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And we deserve a drum roll now.

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X is 100, if we round...

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

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Round your answer to[br]the nearest light years.

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So to the nearest light year

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is going to be 184 light years.

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So X is approximately[br]equal to 184 light years.

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So it would take light 184 years

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to get from Mintaka to Alnitak.

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And so hopefully this actually shows you

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if you are going to do any astronomy,

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the law of cosines, law of sines,

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in fact all of trigonometry,

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becomes quite, quite handy.