Law of Cosines
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Law of Cosines

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!

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