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CCSS.Math: ,

artemus 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 Omnitech 736 lightyears and to Mintaka 915 light-years which are the end points of Orion's belt she knows the angle between these stars in the sky is 3 degrees what is the width of Orion's belt that is what is the distance between Alnitak and Mintaka and they want us the answer to give us a answer in lightyears so let's 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's house right over here this is Artemis's house I'll say that's a 4 Artemis's house and then I'd like to say H let me say this is home this is home right over here and we have these two stars so she's looking out into the night sky and she sees these stars all on the tack which is 736 light-years away and obviously I'm not going to draw this to scale so let's this is all Natak all the tack and Mintaka so this is let's say this is Mintaka right over here min taka and we know a few things we know that this distance between her home and Allen attack is 736 lightyears so this distance right over here that right over there everything will do is in lightyears that's 7 36 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 her Mintaka to her house so this is 915 light years and what we want to do is figure out the width of Orion's belt which is the distance between all Natak and Mintaka so we need to figure out we need to figure out this distance right over here and the one thing that they did give us that they did give us is this angle they did give us that angle right over there they said 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 and 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 to figure out the third side so the law of cosines so let's just let's just apply it so the law of cosines tells us tells us that x squared is going to be equal to the sum of the squares of the other two side so it's going to be equal to 736 squared plus 915 squared plus 915 squared minus -2 times 736 736 x 915 times the cosine times the cosine of this angle times the cosine of three degrees 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 have to cough off-camera because I had some peanuts in my throat was Drive 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 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 a 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 and or another way we could say think about it is X X let me write that X is equal to the square root of all of this stuff so I can just copy and paste that so copy and paste X is going to be equal to this square root of that and so let's get our calculator to calculate it and let me verify that I'm in degree mode is I am indeed in degree mode and so let's exit that and so I want to calculate the square root of 736 squared plus 915 squared minus 2 times 7 36 times 915 times cosine of 3 degrees and we deserve a drum roll now X is a hundred if we round let's see what do they want us to do round your answer to the nearest Lightyear so to the nearest Lightyear it's going to be 180 four light years so X X is approximately equal to 180 for light light-years so it would take light 184 years to get from Mintaka to all Natak and so hopefully this actually shows you it's pretty if you are going to do any astronomy the law of cosines law of sines and in fact all of trigonometry becomes quite quite handy