# Law of Cosines

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BAM!!!, Mr. Tarrou. We just got done learning

Law of Sines and now we are going to learn another law that allows you to work oblique

triangles or non-right triangles. That is the Law of Cosine. Now we need both of these

laws because you can’t always do Law of Sines and you can’t always do Law of Cosine. To

be able to set up the law of sine, you need to have a matching pair of angle and side.

When you have that paring like you know angle A and side a you can set up the law of sine.

However, if you don’t have that then hopefully the law of cosine will allow you to finish

or solve that oblique triangle…find all the missing angles and sides and possibly

even area with a different formula. So we have the law of cosine. a squared equals b

squared plus c squared minus 2bc times the cosine of A. There is three formulas or three

forms of this equation but as you can see from the pattern, they are all pretty much

the same. They are exactly the same. The letters just move around. Whatever side you are setting

the equation equal to, that is what corresponding angle you are going to use in the other side

of the equation in the cosine function. Then everything else the something squared plus

something squared minus 2 times those same two somethings, that is just whatever is left.

So if you are using side b and angle B, then everywhere else in this equation you are talking

about sides of a and c. So there is three forms, but really you could just memorize

one of those formulas and you would be just fine. When do you use these? You use these

for non-right triangles, because if they were right triangles why not just go back to SOHCAHTOA!!!

You use these when given two sides and an included angle. That means that you are going

to know two sides and they are going to come together to form a given angle. Or, you are

just given all three sides of the triangle. Now these are both…were both used in Geometry

for proving that two triangles were either similar or congruent. These are congruency

theorems or similarity theorems and that means that when you use law of cosine, if it fits

the pattern that allows you to use the law of cosine, you are guaranteed that there is

only one possible triangle. It is not that sort um…not arbitrary but you know where

you are doing the sign and you have the possibility of having no answer, one answer, or two of

them. To find an angle…you also want to use the law of cosine to find an angle opposite

the largest side since it may be obtuse. Now when you do inverse cosine with your calculator

which can only work with functions, it is going to give you an answer between zero and

pi or zero and 180 degrees. So, if the angle happens to be obtuse like say 97 degrees you

are going to get the right answer. But if you attempted to use law of sine to find the

opposite the largest side and that largest angle happened to be obtuse, the law of sine

would not be able to give that to you. So if your largest angle was 97 degrees, which

is 7 degrees away from 90 or it has a reference angle of 83 degrees, it is going to give you

an answer of instead… I think I just said that wrong. Instead of giving you an answer

of 97 degrees, law of sine is going to give you answer of 83 degrees. This is because

inverse sine cannot give you answers beyond negative pi over 2 or negative 90 and positive

90. It cannot go beyond quadrant one in a positive direction. So again, if that angle

is 97 degrees and you try to do law of sine you would actually get 83 degrees from your

calculator which will be incorrect. When you want to find that largest angle it is always

suggested to use law of cosine to find that incase it is obtuse, you will get the correct

answer from your calculator. Here we have a couple of measurements and again we are

kind of just basically doing geometry here. We are just working with shapes which means

that as far as I am concerned we should always have a picture. So, I am not going to worry

about drawing this to scale. I just want a place where I can put my numbers down. So

triangle ABC. So angle C is 49 degrees. Side b is 7 units long. Side a is 6 units. Do you

see how these sides that are giving are coming together to form angle of 49 degrees? That

is what we were talking about here with 2 sides and an included angle. The angle is

between the two sides that are given. Now again law of sine would not be possible for

this question because there is no way finding this side right off the bat without doing

law of sine or cosine. You can’t find these angle measures here because, well we only

have one angle out of 180 degrees that are inside so we can’t immediately or easily find

angle A or angle B. So there is no way to set up that law of sine. We don’t have that

side and angle pairing. The only way to do this is with the law of cosine. So that being

said, let’s set that up. So side c squared, we are looking on this one if you want to

look at where the letters exactly where they should be in the formula. c squared is equal

to 7 squared plus 6 squared minus 2 times 7 times 6 times the cosine of the included

angle which is 49 degrees. Now do make sure that your graphing calculator or scientific

calculator is degree mode and not radian mode because you will get the wrong answer even

if you have set it up correctly. Now I can do 7 squared is 49 and 6 squared is 36, and

2 times 7 if 14 and 14 times 6 is um…84. But we are going to have to type this into

our calculator anyway, so if you have a graphing calculator or a 2 line scientific you are

just going to type this in all at once anyway and you get 29.9. Now that is not the length

of c, that is c squared. Don’t forget we are going to have to square root both sides of

this equation. When we square root both sides we are going to get c, the third side, is

the square root of 29.9 which is right off the top of my head approximately 5.5. So here

we go. Now, we need to find…we are going to find all the missing parts. So we need

to find angle A and angle B. Now just to review with you again I am going to do law of sine

for one and law of cosine for the other. That is really not necessary because once I find

one of those two unknown angles i can just subtract from 180 to find the third. But,

to review finding angle measures with sine and finding angle measures with cosine I am

going to both the long way. I am running out of room here so I am going to get this stuff

erased. Let’s find angle B. Now angle B is opposite the largest side of 7, so I am going

to make sure that I use the law of cosine incase angle B happens to be obtuse. I don’t

think is going to be because these are so similar in size, but I am going to do law

of cosine to find angle B just in case it is obtuse I don’t want to get the wrong answer

from my calculator trying to do law of sine. We have…let’s see here. 7 squared is equals

5.5 squared plus 6 squared minus 2 times 5.5 times 6 times the cosine of the angle we are

trying to find which is angle B. I am going ahead and move everything over step by step

so you see what I am doing, and then I just going to pull the answer from say a calculator

or off my tablet I am using to quicken up the pace here. I am also going to keep all

this in exact form. This 5.5 not attached to anything, it is just a positive 5.5 squared,

this is a positive 6 squared, so those are going to get moved using subtraction. So it

is going to be 7 squared minus 5.5 squared, minus 6 squared. Now the 2, the 5.5, and the

6, these are all attached to the cosine of B. This is all one term because they are all

being multiplied together. So that is going to have to be divided away from the cosine

of B. A lot of students will try and put all of these numerical values together, but if

they do that they are not following the order of operations. You cannot add and subtract

before you, these are all touching, so you can’t do all of that before you multiply.

So that is why I moved the 5.5 and the 6 over individually. Now I am going to divide everything

by, or both sides of this equation, by negative 2, 5.5, and 6 all multiplied together. So

negative 2 times 5.5 times 6 equals the cosine of B. Now when you put this in your calculator

all at once or in pieces it should come out to be .261 equals the cosine of B. Now the

cosine function has to get moved away from B. What is the inverse of the cosine function,

how do you undo the cosine function? You do the inverse cosine function. So B is equal

to the inverse cosine of .261. Put that in your calculator and make sure it is in degree

mode. You are going to get 74.9 degrees. So angle B is 74.9. This is our largest side

so this should be our largest angle. If it is not then we have made a mistake. That turned

out to be acute so we could have used the law of sine, but again we did not know that

ahead of time. Now to find angle A we could simply subtract these two angles from 180

degrees. That is fine when you are taking a test and it is timed. You don’t want to

run out of time. But if 74.9 is incorrect, then I am not going that and it going to carry

over to my other answer. So I am going to do the law of sine and just to review with

you as well the other law. And also if they add up to 180, you know there may be some

round off error then we have probably done all of our work correctly. What would this

look like? How would you find an angle measure again using the law of sine. You need a matching

side and corresponding angle. We are going to have, let’s see here, 7 over the sine of

74.9 equals 6 over the sine of A. Our variable is in the denominator and you cannot solve

for a variable when it is in the denominator. I am going to cross multiply this equation…multiply

both sides by the sine of A and multiply both sides by the sine of 74.9 to cancel out that

division. So we have 7 times the sine of A equals 6 times the sine of 74.9. We need to

divide both sides by 7. We have the sine of A is equal to 6 times the sine of 74.9 degrees

divided by 7. That means the sine of A is approximately…well that is what happens

when you do things off the top of your head. I picked a different number so I will be right

back with a calculator. Ok, so…get this thing turned on. Current document. So I need

6 sine of 74.9 divided by 7 comes out to be .827… actually .828 correctly rounded off.

Then A is equal to the inverse sine of .828. So A comes out to be the inverse sine of .828

which comes out to be approximately 55.9. Since I have gone off script here a little

bit. Let’s make sure these do add up to roughly 180 degrees. So 55.9 plus 49 plus 74.9 and

it is 179.8. So that .8 of a decimal is simply because of a little bit of round off error

I have in there. Again, you want to… You can find any measure you want both sides or

angles using law of cosine. I guess not all the time but occasionally. But when you are

looking for those missing angle measures you do often have a choice, but please do not

try to find the angle opposite the largest side using the law of sine. If you do, you

could possible get an incorrect answer. Again if that angle happened to be 97 degrees, the

law of sine would come out to be 83. Not probably…it would. Don’t use law of sine to find the largest

angle. I am Mr. Tarrou. Go Do Your Homework!

ProfRobBobPost authorYou make a teacher proud:D…see that big smile on my face!

And if you are still studying 9 hours later, I know your going to be ready to go into class and show your teacher who's been studying and who hasn't!!!

Wahoo GonzalesPost authorDude when you help me you give me enough knowledge that i get 80s and 100s

ProfRobBobPost authorYou know it takes dedication from you to make that happen…I'm just here to furnish you with the lessons:)

Keep up the great study habits and you'll change those 80's to 90's:)

I just reminded you in another comment to keep sharing my channel with everyone:D Thanks for supporting!

Alexandria CostaPost authorI'm just letting you know that I got a 93% on my final exam in Honors PreCalculus!! Your videos really helped me and they're the main reason I got an A. Thank you so much and I'm definitely going to use your videos for AP Calculus next year!

Christian GarciaPost authorWhat do I do if no angles are given?

ProfRobBobPost authorStart the problem and find the angle opposite the largest side in case it is an obtuse angle. You would start with work like I am doing starting around minute 7:30.

FolderolPost authorMe: Why do I keep getting decimals?

ProfRobBob: …and make sure it's in degree mode.

Me: …

Me: -facepalm-

ProfRobBobPost authorI do that too sometimes. I am reposting a very long video to fix a mistake where I said 1+2 is 2, so it happens to all of us. I am glad I reminded you to check your calculator:)

Alexandria CostaPost authorI can't remember if I commented back, but I owe you a lot of thanks because I studied for my Honors Pre-Calculus Final solely by watching your videos and I got a 93! So thank you!:)

ProfRobBobPost authorYou did comment back…I just saw it when I came to this page to send you a reply today 7/22/13. I don't how I missed it, other than we did have some YouTube gmail issues:(

Awesome news on your "A"…sounds like together we make quite a team and I look forward to helping you with AP Calc next year!…which is sadly, right around the corner:(…where did the summer go?!

Did you see me on the NBC nightly news with Brian Williams 7/1/13?..check it out if you're bored.

ProfRobBobPost authorPS…THANK YOU for adding my channel to your :)CHECK OUT list as promised…I really do appreciate the support of loyal viewers like yourself!!!

Mackenzie CrossPost authorYou are seriously helping me majorly! I'm taking this course online and my grade at the end of the term determines if I get to go back to school or not. Up until now I thought I couldn't understand any of it and that I was doomed to fail! Now, I get everything you're saying and I feel like I grasp the concept. You are a great teacher!!!!

ProfRobBobPost authorThank you Mackenzie…and you are an awesome student for not giving up and reaching out for some extra help! I hope my videos and your new found confidence continue to help you pass your on-line course with ease. I look forward to hearing the good news regarding your return to school…keep working hard and aim for that success thats right around the corner:)

Squiffy FartPost authornot just sohcahtoa but SOHCAHTOA!!!

Morgan RockettPost authorThis was definitely comprehensive- thanks a lot.

ProfRobBobPost authorWhat can I say…Math brings the geek out in me:)

ProfRobBobPost authorYou're welcome…Thanks for choosing my channel to learn form:)

SU YAPost authorhellow can u help me the formula of this given? a=8 b=7 c=? angle A=50degree B=? C=?

ProfRobBobPost authorBecause the information given is in the SSA pattern you want to watch my video Ambiguous Case Law of Sines. Thank you very much for watching.

Paul ElkerPost authorNot that it really matters, but at the 5:30 mark you mention side a as 7, when really it is 6. You still get the same answer but I just wanted to bring it to your attention.

ProfRobBobPost authorThank you. Addition and Multiplication is commutative so I don't worry too much about exactly following the pattern of the formula. But, when doing an example I should have been a bit more careful about that:) Thank you and thank you for watching!

Stefan HerzogPost authorThank you so much for making videos I have found that as a student in college I really can't rely on my proff for teaching me the material thoroughly enough, so I am really glad that I have these videos to prepare for exams.

Thank you so much you are a really good teacher.

ProfRobBobPost authorI approve all my comments to keep the channel suitable for all audiences and I just answered your first comment…I'm glad to see you are still watching more lessons and I hope you continue to find them helpful.

I've had many viewers tell me they have started study groups and all watch a video and then together they work out problems…it always seems less dreadful if you have people to share with:)

surya narayananPost authorI Like Your Jumping ðŸ˜›

ProfRobBobPost authorBAM!!!

…and "Thanks" for watching:)

Lino GovaPost authorthis is great, i had to miss class today so this more than made up for what i miss. Thank you!

ProfRobBobPost authorThanks for watching…I hope you will return whenever you need more math help in the future:)

Lino GovaPost authoryes of course, you got a new sub.

ProfRobBobPost authorTHANKS for subscribing and supporting!

Jay BayPost authorBAM!

MikePost authorGood video. Helped a lot. Â I like how randomly energetic you became on certain things haha.Â

Dylan JenkinsPost authorYou're the bomb man. SOHCAHTOA!!Â

ZaeemPost authorThank you! Great teacher, always come to your videos help help.Â

Keep doing what you do!

Karen YangPost authorthen… can you use law of sines within an obtuse triangle to find one of its acute angles or corresponding side lengths? I know you can't find the obtuse angle/longest side length, but what about the other two acute angles?

os rsPost authorThanks a lot man I'm doing bad in geometry cuz of my confusing teacher but your videos are making up for it

Rihua chenPost authorThanks for adding in different scenarios like theÂ bearings and whatnot. Great guide to review from!

CoffeeJunkiePost authorCouldn't you just use sinA/a = SinB/b?

xXItstrayceceXxPost authorThank you so much! I was so confused in math class untill I saw this video! I really appreciate it! ðŸ™‚

UnleashedMonkeyPost authorgreat video, really helped!! small suggestion: maybe include a sample problem in the video description(with the answers all the way in the bottom) for viewers to try and test it out. by the way, I subbed. I'm sure I'll be on this channel a lot throughout the semester ^-^Â

Cedric NkollaPost authorYou are the best! I can even skip class and still get an A in my Precal class thanks to your excellence in Pedagogy!

Iran ArellanoPost authorWait isnt a supposed to be 6 and b supposed to be 7?

MikailaPost authorWhen you say SOHCAHTOA during the first 2 minutes i actually cry!! omg your to greatt! so helpful!!Â

josh futureheartPost authorDo you have a video about Law of Tangents?

Aryanne_v2Post authorThe Law of Cosines and Sines confused me so much. Thank you for these. They explained them perfectly.

Trieu NguyenPost authorlove your lesson keep it up. thank for the help

Joshua CelisPost authorYou are the reason I think I will pass my final tomorrow. Thank you keep working hard.Â

jzim123Post authorI plugged in the law of cosines just like you did, but can't figure out how you get square root of 29.9 for side C. I keep getting c=(cos 49), and getting a very small number for side C.Â

John Carlo EstebanPost authorBAAAM! Thank You so much for this Sir! ðŸ™‚

Darien Perez-MartinPost authorIt would be awesome to see some applications. Off the top of my head I could think of how useful this would be to calculate the distance between various test charges and a source, and then us the results to calculate the strength of the electric force at that location in the electric field! Thank You!!

Joe WPost authorThanks, Professor. Great lesson!

Giovanny MontezPost authorThanks!!

BoomPnaesPost authorSir i have a question why my teacher says that the correct formula for finding angle for law of cosine is b^2+c^2-a^2 divided by 2(b)(c) if finding the cosineA like that but why youre solution is…… b^2-c^2-a^2 divided by -2(a)(b) why its negative? i really confused sir Bob if which is true, i hope you can read what i post, thank you so much. im your #1 subscriber.

Charles HenryPost author4th video about laws of sine and cosine; and the other's were not nearly as informative/helpful. Â especially about the law of sine not working for obtuse angle.

Justin DixonPost authorAlright… whoa whoa WHOA… your first example says 7^2=5.5^2+6^2-2(5.5)(6)Cos(B)

That makes no sense because it was a C^2= problem, but C = 5.5 according to what you've got listed on the top of the board, A=6, B=7… none of that makes any sense.Â

Kelvin LaoPost authoryou are the man. Without you, i would've definitely failed.

Billy JamesPost authorThank you so much! I don't know what I was doing wrong but I was just not getting it. I watched a couple videos and looked up step-by-step solutions and yours is the only one that actually made sense! Thank you! ðŸ™‚

Pranav VinchurkarPost authorThanks! Really helpful.

MerlinPost authorI cant get the .261. How did you get that. Even if put it all into my calculator.

stefranqui C.Post authorAwesome awesome awesome!!! ðŸ˜€

Inan IsmailovPost authorYou have no idea how happy i get when you jump in the video and will get the material in my head in 15 minutes.

Î›ÎžÐ¯ÐžÐ˜Î›Ð¦Î¤ Î›ÎžÐ¯ÐžSÎ¡Î›Ð¡ÎžPost authorHi Professor! ðŸ™‚

I really appreciate your instructional video and it's more clearer to understand. ðŸ™‚

And actually, on the spot, I quickly understand the lesson.

Thank you so much for helping not only me but other students which are not good in Math. ðŸ™‚

And actually now is just 1 hour before our quiz when I watched this. ðŸ™‚

nile majesty lobustaPost authorjust have been doin' my 100-item assignment for almost 2 days, and when I've got your tutorial prof, so happy that i just have to download it and make it as my tutor.. thank you so much..

MrSaintskid11Post authorThanks So much… I would not have passed my test without this!

Jared atienzaPost authorGOOD VIDEOS. Hope i could pass my exam with your help!Â

GutierrezAnthonyPost authorHonestly can't thank you enough for really breaking this down for me to fully comprehend, gonna ace this test in honor of you.

Ken BonillaPost authorOkay, so why when you have found the values of sides a,b,c, don't you just use the law of sines to find the smaller of the two missing angles, and then just subtract the two known angles from 180 for the remaining angle? Seems like a lot less work, and since there can never be more than one obtuse angle there isn't any way to mess up.

ProfRobBobPost authorLaw of Cosines is now Closed Captioned:D

Rob GainesPost authorWOW THANKYOU SO MUCH

Sunday DammyPost authorThis is awesome! Explain better than my professor at school. Thank you so much sir! You just made my day… May God continue to bless you abundantly!

Wubs4ScrubsPost authorLearned more in 15 minutes than I did during a 90 minute class period, thank you so much!

PuresynergyfitnessPost authorwish you also had a site with worksheets / practice problems and a key

anmol choudharyPost authorthis video confused me

Juan IgnacioPost authorAnother great video! Thanks!!

ShamamalamaSamaPost authorI love you.

theBoiiiiiiiCRemePost authoryou're awesome! You made understanding these topics easy!

Tanisha WilliamsPost authorI love your energy and animation! lol

NadimPost authorIt's awesome, thank you very much!

JansennPost authorProfessor ,about the ending of the video , can I use this formula in order to get Angle A?

A = 180-(B+C) ??

btw sir, you're awesome ðŸ˜€

Andi DittakaviPost authorI learned so much more by watching your vids thank you so much!!!!!!!!

Anthony Wesley SilvaPost authorHe contradicted himself when solving for angle A. Instead of using b/SinB he should've used the given information which is c/SinC. If a mistake was made it would've carried over using b/SinB=a/SinA

Brandon HodgesPost authorambiguousÂ angles are what you are thinking of lol

Branco Sebastian MartinezPost authorif only i knew how to use the calculator

JeromePost authorLaw of tangents pls

Sheila OnaPost author7^2=5.5^2+6^2

it should be + not – 6^2

Rae JohnsonPost authorMr. T thank you so much for this video! I appreciate your excellent instruction and great teaching style. You're really a gifted teacher!

Tes HPost authorquestion here…in the last example of this video, we're given two sides and one angle of a triangle, however after applying the law of cosine to find the the third side, can't we determine if the opposite angle of the largest side is obtuse or acute from the Pythagoras theorem and then we can straight head on to the law of sine to find the largest angle and then we take the supplementary angle instead by subtraction from 180Â°?

while # 1 below is obvious, I'm not sure if # 2 and 3 will make any sense or if they're already established rules please excuse my slacking for not researching but right off my head I felt that,

1. right triangle, if c^2 = a^2 + b^2

2. if c^2 < a^2 + b^2, then the opposite angle of the largest side would be acute

3. if c^2 > a^2 + b^2, then the opposite angle of the largest side would be obtuse

Tyler BhamePost authoryou saved my grade thank you!

Sameerah aPost authorReally helpful but could you have used sine law to find the rest of the angle ?

vylon10Post authormy college professor asked me to discuss law cosines in class, this video will really be a big help thanks

justforyouenjoiPost authorThanks for the video. It really helps a lot. My school uses pearson online with Tom Atwater who is the worst video instructor of all time.

Luis abcdefgPost authorHe mixed a^2 and c^2. I thought I was going crazy.

dudee wtfPost authorAmazing video!!! Thank you

Brahmini RaviPost authorI hate trig

Kenzo TenmaPost authoryou're awesome man I like your videos, I learned a lot honestly. ðŸ˜€

ivanPost authorBam what a lesson prof bob

Nicolas NepomucenoPost authorwhen your teacher is crap so you have to resort to youtube for help.

Joey LynnPost authorTell me why heâ€™s more helpful than my precal teacher.

Luis DiazPost authorI could picture you jumping when you are 80 years old. BAAM there goes AlGEbra.

Tanya ToddPost authorIs that an old school pencil sharpener to the right?? lol GREAT VIDEO! Thanks!