BAM! Mr. Tarrou. In this video we're gonna take a look at two examples of solving
for the missing parts of oblique triangles using either law of cosine or
law of sine actually it's gonna be a lot of law of cosine because that is what
the point of this video is to introduce the law of cosine in the description
you'll find a link to my video introducing the law of sine and a link
to my web page that has all my old videos for law of sine cosine
applications of law of cosine like barring problems and such and this time
we're going to actually try and make this video without a sign errors so
thank you very much Brian for pointing that out
at least this original video was only up for about ten hours before someone got
my sign error okay so we're going to do a little bit of review to begin with
first of all in Part A we have that was good catch we have a triangle where we
have all three sides but we don't know any of the measure of the angles so this
information is given to us in a
side-side-side pattern that's a congruence theorem that you learned in
geometry which means that these three sides I mean if I'm giving you these
three sides or kind of assumed to make a triangle I mean it's not like it's a
length of six and one that don't even reach to a length of eight and since
that's a congruence theorem one triangle we don't have like an ambiguous case
like we might have sometimes using when we need to start our problem with the
law of sine and we have another problem here of side B is two side C is 90 s
excuse me angle C is 96 degrees inside a is three and it's never a good idea to
start a geometry problem without well this is effective arguably trig but it's
applied to sort of an geometry situation it's not good to start these without
having a picture so I'm gonna just draw a triangle not be considered concerned
about drawing it to scale just something where I can say oh let's let this be
angle a this be angle B this be angle C side B which is opposite angle B is a
length of has a length of two side a opposite of angle a has a length of 3
and angle C is 96 degrees and we're doing
this now also to identify do we have information given to us in the pattern
of a congruence theorem from geometry which we do side-angle-side and this is
going to help us in a couple of ways it's going to help us determine whether
we start these problems with law of sign or law of cosine and it's also going to
be a reminder as to whether we have just you know a triangle that we're solving
for or if we have an ambiguous situation where it's side-side-angle
spell that backwards it's not a good word right side-side-angle is not a
congruence theorem in geometry which opens the door for us to not even
possibly have a solution or actually have two triangles that we need to solve
for well as we spoke about in our previous lesson for applying or starting
these problems with the law of sine we have to know information about a side
and the angle opposite to that side and we don't have that situation in either
one of these scenarios so both of these need to be started with the law of
cosine okay so that is the we have a side squared is equal to the other two
sides squared and added together -2 times those other two sides times the
cosine of a your textbook is probably going to be given you three basically
three equations for the law of cosine I say this because they just have one
where it's set equal to a squared one set equal to B squared once at equal to
C squared instead of remembering three formulas for the law of cosine a side
squared not the letter A but just a particular side squared is equal to the
other two sides squared added together minus 2 times those other two sides
times the cosine of the angle which is opposite the side that it's basically
set up to equal not exactly equal to the side but the side squared your textbook
probably also gives you a version for the law of cosine identity your formulas
that are set equal to the cosine of the angle like cosine of a cosine of B
cosine of C I would suggest that you don't necessarily try to memorize those
it's just Y remember basically six formulas when you can kind
of just remember one another reason is they move these terms of well in this
case B squared and C squared C squared over with subtraction divided by
negative 2b and then for some reason they always simplify the denominator and
instead of showing it as a division of negative 2bc
they showed as a division of positive to BC that means you have to apply a
multiplication of negative one through the numerator and it just it messes up
the signs and a lot of times when kids try to memorize that version of the
formula they make a sign error and they miss a problem they shouldn't otherwise
miss now with all of that being said one more thing one more review caveat piece
of information before we get started we did not have to find any angle measures
for our oblique triangles in our first lesson because we were using the law of
sine and the information was given such that we just didn't need to when you
start trying to find missing angles and oblique triangles you need to be really
careful because well the inverse sine of
Y over R or opposite over hypotenuse but we like Y over R and the labels of XY in
R when we're dealing with angles in standard position is going to give you
some kind of angle measure and we need to remember that that angle measure that
we get out of the inverse sine function is going to be between the measurements
of well theta is greater than or lum greater than or equal to negative 90
degrees and less than or equal to positive 90 degrees 99 or negative PI
over two and positive PI over two if you're working with radians which means
that you cannot find the measure of an obtuse angle with inverse sine if you
are going to be looking for this angle C and this angle happens to be 93 degrees
and you try and find it with inverse sine you're going to be because sine is
an even function instead of being told that the
obtuse say 93 degrees the inverse sine function is going to tell you that it's
87 degrees where 93 and 87 degrees are both either three degrees off the y-axis
but that doesn't have like a really special name or they both have a
reference angle of 87 degrees that'll throw you off you won't know that you've
gotten the wrong answer so never ever use law of sine to find an angle measure
that's opposite the largest side of a triangle if it's such that that angle
was supposed to be obtuse you would get an acute answer and continue the problem
without ever realizing that you had made an error so it's technically sort of not
an error it's just a limitation of inverse sine now when you do inverse
cosine the angle you get from that is going to be an angle that is between 0
and 180 degrees so the equals obtuse you're going to get it like we're about
to do right now we're gonna start here with saying that 8 squared is equal to
the other two sides squared and added together so 6 squared plus 5 squared
minus 2 times 6 times 5 times the cosine of C now we're going to bring the 6
squared and the 5 squared over to the left-hand side with subtraction we're
going to divide both sides of this equation by basically negative 60 right
through 6 times 5 is 30 times 2 is negative 60 now here's where I flubbed
up my first lesson and made a sign error we have 8 we have 8 squared which is 64
minus 6 squared which is 36 minus 5 squared which is 25
I even reread my work but because I did it you kind of don't catch it as well we
have 64 minus 30 is what 34 now 34 minus 20 is 14 we've got 14 minus 6 which is 8
and we have 8 minus 5 which is equal to three not negative
three like I wrote in my initial recording of this video and then we have
that divided by negative 60 so we're looking at negative one twentieth and
let's go ahead and find out what C is by applying the inverse cosine function to
both sides so C is equal to the arc cosine or the inverse cosine depending
on the book you're using means the same thing and that is going to come out to
be just I'm typing it in correctly ninety two point eight let's round it
off to the nearest tenth ninety two point nine degrees now see if I found
that later using the law of sine I would get
basically about 87 degrees not the correct measure angle of ninety two
point nine degrees now as Brian pointed out I was teaching by example I told you
if possible don't ever do another part to your problem with information that
you found on your own because if you do you're going to carry that error on to
the next problem well piece of information so when we go
to find angle a if I used the law of cosine and for some reason did something
silly like wrote negative three instead of positive three instead of negative
three here if that error is if that angle now is wrong because of a silly
mistake I'm going to get the wrong answer for angle a so if I don't want to
take an error and carry it forward I can well just start fresh again with
the given information and say that we have five we're going to review law of
sine in the next problem though by the way so don't worry about that or in the
blue chalk code we have five squared is equal to six squared
ma plus eight squared minus two times six times eight times the cosine of a
and same thing we're going to have five squared minus 6 squared minus
squared over negative 2 times 6 times 8 is equal to the cosine of a and another
cautionary note with these problems a
little bit of roundoff error can affect and spiral into larger and larger sort
of small errors becoming large errors as you work through the multiple parts of
your triangle it's really best I mean we're going to yeah I can do 25 minus 36
minus 64 in my head at least that's what I thought yesterday at any rate but I'm
gonna have to grab the calculator at some point regardless and especially now
that we have all these fancy 84 even like a black and white 84 back in my day
you had to get all that syntax super correct you can type this in your
calculator exactly how you're writing it and then you don't have any roundoff
error like don't take this decimal like this numeric this numerator and then
divide by the denominator and round that and then do the input the inverse cosine
and around that and maybe if there's another step around that it can throw
off your answer by 1 2 3 4 5 degrees when you really you're doing everything
just perfectly fine so don't do that second cosine alpha y equals to open up
the fraction menu and here we go and we get a is angle a the measure of angle a
is equal to thirty eight point six degrees now for remaining angle of
course we know that we have to have measurements that add up to 180 degrees
so we have 180 minus ninety two point nine minus thirty eight point six let's
just not grab our calculator for every single little piece of arithmetic we
have to do like in universities they don't even allow calculators in calculus
we have 180 minus 90 is ninety we have ninety minus 30 which is sixty now we're
down to what sixty minus two is 58 now we're down to 50
now we're down to forty nine point one and forty eight point five as we take
away that point six no I don't want to put out another video with a small error
so even though I'm saying practice your arithmetic mental arithmetic we're going
to check fantastic okay now Part B we're going to have to distort this problem
just because the way the information is given find the length of side C first so
once again a side squared is equal to the other two sides squared and added
together so we're gonna have three squared plus two squared minus two times
those other two sides times the cosine of the angle measure that angle that's
opposite the side that you're working with so the cosine of ninety six degrees
and that's going to be C squared is equal to we've got again I just said we
could type all this in the calculator but let's go ahead and just walk through
this we have nine plus four which is 13 minus 6 times two which is 12 and C is
then equal to the square root of this and that is equal to three point eight
units double check three point seven eight three point eight fantastic we're
gonna go to the nearest tenths place now now that we know that the length of C is
equal to approximately 3.8 let's do a review of law of sine so we are going to
find
the measure of let's say angle B using the law of sine because now we know a
side and the measure of the angle opposite that side we also know that
angle B is not opposite the largest side of the triangle therefore there's no
possibility for it to be an obtuse angle so we have and because we're looking
forward an angle measure I'm going to put the sine of B on top so we have the
sine of B over 2 is equal to the sine of 96 degrees over 3.8 so we're going to
multiply both sides by 2 and have the sine of B that's 2 sine of 96 degrees
over 3.8 take the inverse sine of both sides and the measure of angle B is
equal to the arc sine I want to write this just in case your book is using it
this is the same as inverse sine 2 times the sine of 96 degrees over 3.8 I keep
using the degree notation because you will be eventually working with radians
if you have not already and that's how you tell whether or not you're working
with a degree measure or a Radian measure use the degree notation and that
comes out to be thirty-one point six degrees and one more time for angle a
then the measure of angle a is going to be well the result of doing 180 minus
thirty-one point six minus 96 through this in our head one more time we have
180 minus 90 is ninety degrees now we're down to 90 minus 30 is 60 59 15
- six is what 53 and now 53 - point six is fifty two point four degrees and that
is how you hope you don't enjoy your introduction to the law of cosine one
more time links to old videos and application for law of sine and cosine
or in the description I miss a true hey go to your homework
you