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# Solving for an angle with the law of cosines

CCSS.Math:

## Video transcript

let's say you're studying some type of a little hill or rock formation right over here and you're able to figure out the dimensions you know that from this point to this point along the base trade along level ground is sixty meters you know this deeper side steeper I guess surface or edge of this cliff or whatever you want to call it is twenty meters and then the longer side here I guess the less deep side is 50 meters long so you're able to measure that but now what you want to do is use your knowledge of trigonometry given this information to figure out how steep is decide what is the actual inclination relative to level ground or another way of thinking about it what is this angle theta right over there and I encourage you to pause the video and think about it on your own well it might be ringing a bell well you know you know three sides of a triangle and then we want to figure out an angle and so the thing that jumps out at my head is well maybe the law of cosines could be useful and let me just write out the law let me just write out the law of cosines before we try to apply it to this triangle right over here so the law of cosines tells us that C squared is equal to a squared plus B squared minus two a B times the cosine of theta and just to remind ourselves what the a B's and C's are C is the side that's opposite the angle theta so if I were to draw an arbitrary triangle right over here and if this is our angle theta then this determines that C is that side and then a and B could be either of these two sides so a could be that one and B could be that one or the other way around as you could see a and B essentially have the same role in this formula right over here this could be B or this could be a so what we want to do is somehow relate this angle we want we want to figure out what theta is in our little hill example right over here so this is going to be theta what is C going to be well C is going to be this 20 meter side and then we could set either one of these to be a or B we could say that this a is 50 meters and B is 60 meters and now we can just apply the law of cosines so law of cosines tells us that 20 squared is equal to a squared so that's 50 squared plus B squared plus 60 squared minus 2 minus 2 times a B so minus 2 times 50 times 60 times 60 times the cosine of theta times the cosine of theta this works out well for us because they've given us everything there's only really only one unknown there's theta here so let's see if we can solve for theta so 20 squared that is 450 squared 50 squared is is 2560 squared is 3600 and then 50 times 6 we'll see 2 times 50 is 100 times 60 this is all equal to 6,000 so let's see if we simplify this a little bit we're going to get 400 is equal to 2500 plus 3600 let's see that'd be 60 100 it's equal to 6,000 let me just in a new color so when I add when I add these two I get 6100 I do that right yeah so it's 2,000 plus three thousand five thousand five hundred plus 600 11 hundred so I get 60 100 minus 6,000 minus six thousand times the cosine of theta times the cosine of theta and let's see now we can subtract 60 100 from both sides so I'm just going to subtract 60 100 from both sides so that I get closer to isolating the theta so let's do that so this is going to be this is going to be negative 5,700 is that right five thousand seven hundred plus yes that is right right because if this was the other way around if this was six thousand one hundred minus 400 it would be positive five thousand seven hundred all right and then these two of course cancel out and this is going to be equal to negative six thousand times the cosine of theta times the cosine of theta now we can divide both sides by negative six thousand by negative six thousand negative six thousand and we get I'm just going to swap the sides we get cosine of theta is equal to let's see we could divide the numerator and the denominator by essentially negative one hundred so these are both is going to become positive so cosine of theta is equal to 57 it is equal to 57 over 60 and actually that's can be simplified even more three goes into 57 is that nineteen times yep so this is actually this could be simplified this is equal to nineteen nineteen over twenty and we actually didn't have to do that simplification step because we're about to use our calculators but that makes the math a little bit our little it makes it a little bit more tractable right three goes into 57 yet nineteen times and so now we can take the inverse cosine of both sides so we could get both eight ax is equal to the inverse cosine with the arc cosine of nineteen over twenty of nineteen over nineteen over twenty so let's get our calculator out and see if we get something that makes sense so we want to do the inverse cosine of nineteen over twenty nineteen over twenty nineteen over twenty and we deserve a drumroll we get eighteen point one nine degrees and I already verified that my calculator is in degree mode so it gets eighteen point one nine degrees so if we want it around this is approximately approximately equal to eighteen point two degrees if we want to round to the nearest tenth so that's essentially gives us a sense of how steep this slow actually is