BAM!!! Mr. Tarrou. We are going to take a look at over the next few videos a couple
of laws, we are going to lay down the law, that allow you to solve for missing sides
and angles of non right triangles, oblique triangles. Now the first one that we are going
to take a look at is Law of Sine which says... Let's not forget that lower case letters are
for sides and upper case letters are for angles. So we have side a, or a over the sine of A
equals b over the sine of B equals c over the sine of C. All of the lower case letters
are sides. Now we can't have an equation with two equal signs in it at the same time. There
is no such thing as a, well I am not going to work with a three sided equation. We are
only going to use two of these at a time. We are going to know, we need to know an angle
and it's opposing side. We need at least that set up as enough information to start the
problem and then we will set up another fraction based on whatever missing parts of the triangle
that we are dealing with. Now not all oblique triangles can be set up and solved using the
Law of Sines. If you cannot use the Law of Sines to solve a triangle, to find it's missing
angles and sides you can use the Law of Cosines. This is in another video. I will put a link
to it above my head. So before we get started though, there is another video lesson after
this as well that is going to be the ambiguous case for law of sines. See, you are just going
to be given information about a triangle and you are going to need understand if that information
given sets up one triangle, maybe it does not even set up a single triangle at all,
or there is two answers. That is the ambiguous case for law of sines. So as we look at these
problems today, we need to keep in mind some Geometry postulates or theorems that you learned
a couple of years ago. The congruency theorems you learned for proving that two triangles
we congruent, or the same size and shape. Similarity theorems as well but we are going
to leave out the AA postulate. So your congruency theorems from Geometry, some of them were
angle-angle-side, side-angle-side, angle-side-angle, and side-side-side. There was a hypotenuse-leg
theorem as well if you were dealing with a right triangle, but we are not. We are not
going to be proving that two triangles are congruent. We are not doing two column proofs.
But we are going to have to look at the information given to us and see if it follows one of those
four patterns. I am also not saying that all four of these patterns are set up for the
law of sine either. Some of them will be for the law of cosine. Like for this side-side-side
set up, the only way to solve for that oblique triangle would be with law of cosine which
is in a later video. It is a topic that is probably in the next section of your textbook.
What we want to pay attention to is that we do not have a set up like this. Again, I am
just giving you a preview for the next video, the ambiguous case for law of sine. If information
is given to us in the SSA pattern, and we all know what that spells backwards. It is
not a good word because we can't use it. That is not a congruency postulate, but some questions
will give you information in this setup. If you have that, then there is a possible of
no triangle being set up, you cannot even do one at all, or one answer or two answers.
So thus, the ambiguous law of sines follows this postulate of SSA, not postulate but pattern.
It is not a congruency theorem. When you get this, you know that you have to a much tougher
question. It might not be a triangle at all or you might have two answers. But that is
not for this video, so let's just forget about it for now. Our first example is going to
be a triangle looking like this. We have angles of A, B, and C. I am not drawing this to scale.
Never trust a drawing is to scale unless it says that in a math book. So we have a angle
A of 44 degrees and angle C of 61 degrees. Opposite of angle C we have side c which is
going to be 52 units. Now, as I go around this triangle and I identify how the information
is given to us, if I go in a counter-clockwise rotation and you are not allowed to skip a
side and an angle. I have an angle 61 degrees. No information about the side here of side
b. So I need to have information about the next angle. I have angle, angle, side. AAS
That is a congruency theorem from Geometry so this is only going to have, with information
given, there is only one possible triangle. While this is not drawn to scale, this is
the triangle. Now I cannot go around in the other direction. I can't say it is Side, and
then skip an angle and skip a whole side. I cannot skip two parts of that triangle.
This is not side, and then angle, angle. I could do this this way. But as you try to
look at the information given, go either counter-clockwise or clockwise, but you can skip one side or
one angle that does not have information but not two parts. You cannot rotate around this
way and say that it is side-angle-angle because you are skipping two blank pieces of information.
Ok, let's solve this triangle. Enough talking! So we have got... Let's find side a. So we have
a, lower case a, over the sine of 44 equals 52 over the sine of 61. And here we go. If
you are going to set up the law of sine in an oblique triangle, you need to have a side
and it's corresponding angle paired up. You need to know both of those to be able to work
out the law of sine. If you do not have this opposite paring, then the oblique triangle
is going to need to be set up and you are going to solve it by using the law of cosine.
However we do. We have a divided by the sine of 44 equals 52 over the sine of 61. Any time
we are trying to solve an equation and we want to move something away from the variable,
we always just have to do the opposite math operation. So that is a divided by the sine
of 44, so we are going to multiply both sides by the sine of 44. So we have a equals 52
times the sine of 44 all over the sine of 61. Ok, so we have 52 times the sine of 44
divided by the sine of 61. If you are using a TI-83 or TI-84 with older software, you
can't put it in the calculator in this math type format. You want to wrap the 52 in parenthesis,
when you type in sine it is going to open up another parenthesis for you automatically,
close those parenthesis before you type in the division of the sine of 61. If you get
that into your calculator correctly, you are going to get is approximately 41.3 units.
Ok, now if you are in a science class your science teacher might be really enforcing,
or your math teacher, might be really enforcing the idea of significant digits and they might
not really like you putting calculated values that appear more accurate then the measured
values. Like really is rounded off to the nearest whole number and there is no point
zero, so this technically needs to be rounded off to 41. I don't really do that ever so
this is 41.3. Now we want to find side b. Well to find side b, I have a side angle pairing
but I don't know this side and I don't know this angle. I cannot do law of sine if I have
both of those pieces missing, the angle and the corresponding side. But, there is 180
degrees inside of a triangle so it should not be too hard to figure out. We have 180,
now we have minus 40 is 140, 140 minus 60 is 80, now it is 79 and 75 degrees. Ok, so
we have enough information now to find side b. Ok, so we have enough information now to
find side b. I am going to do b over the sine of 75 degrees equals, and I am going to use
this 52 and 61 degrees pairing again. If I hit a button wrong and the 41.3 is a little
bit off because I made a careless mistake, I don't want carry that mistake into another
answer of the question. So it is going to be b over the sine of 75 degrees equals 52
over the sine of 61 degrees. B is equal to 52 times the sine of 75 degrees divided by
the sine of 61. That comes out to be approximately 57.4 units. One more example, actually two
more. We are going to do one more example with law of sine. Then I want to explain about
finding the area of an oblique triangle using the sine function. So we have this information.
The measure of angle A is 10 degrees. The measure of angle C is 43 degrees. Side b is
equal to 21 units. Well, I don't have a drawing, but we are talking about triangles. We are
talking about shapes. It is kind of like we are doing Geometry. We are doing Geometry.
So any time we are doing Geometry, excuse me, I like to give myself a drawing. Clearly
I did not look at the numbers. I just made a generic triangle. I am not drawing it to
scale. I just want a place to put the numbers, so I can see what is going on. I don't care
about the measurements appearing to be correct. A is ten degrees. That is almost 45 degrees
the way I drew it. Angle C is 43 degrees, and side b is 21 units. Now if I go around
this triangle, my information is given to me in the angle-side-angle pattern. So angle-side-angle
is a congruency theorem we learned in Geometry and thus there is only one solution, one triangle
that you can set up with this given information. I am going to help you set up one side and
not really finish this one because it works exactly the same as the other example. We
are going to find side a. So a over the sine of 10 degrees equals, now I have the 21. But
again you cannot set up the law of sine unless you have a pairing of side and angle. So while
I have 21, and that is the only side of the triangle that I know. If I am going to use
the law of sine, I have to know the angle opposite the 21. But, again the inside of
a triangle must add up to a 180 degrees. So we have 180, now 140, now 130 minus 3 is 127
degrees. Ok, so now I do have that opposite side angle pairing and we get 21 over the
sine of 127. We multiply both sides by the sine of 10 to undo that division and we get
a is equal to 21 times the sine of 10 degrees over the sine of 127. BAM! Ok. Now just type
that into your calculator right and you will have the answer...or it is 4.6. Ok, for our
last example we are going find the area of an oblique triangle. Now you are going to
learn two new formulas for the area of a triangle. I have a video posted up already that ties
both of those together in the same example. But since... You know I am going to go over
this formula for you in a second. It is in probably the same section of your textbook
where you have learned the law of sines. So let's tie in some knowledge that you already
know with these new formulas that we see for the area of a triangle, which are honestly
just convenient for certain cases. We have this triangle. So we have an oblique triangle,
it is not a right triangle. We are going to find the area. Well, the area of a triangle
is equal to one half the base times the height. If you are going to find the area of a triangle
using this formula you have known for years now, one half the base times the height, you
need the length of the base and you need the height. Let's pick a different color. Ok,
so the area of a triangle which is one half the base is nine. We are going to find the
height. Now the height is not seven, the height is not this unknown value here, clearly this
is at a slant. How are we going to find the height of that triangle, that oblique triangle?
I have already kind of given you a hint here, because I have dropped the altitude down to
make a perpendicular connection or a ninety degree angle. So what we are going to do is
to find the height, is we are going to highlight this triangle here. We are going to remember
that the nine is at least for now, not a number that we are interested in because is the entire
length of the triangle which is not a right triangle. But for the green triangle that
I have temporarily drawn over our diagram, we have a right triangle so we can go back
to SOHCAHTOA! Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent
is opposite over adjacent. From the 21, what do we have and what do we want? We want the
opposite side, which is really not a side our triangle, but is the height. We have the
hypotenuse again of the green triangle. So the sine of 21 degrees is equal to the opposite
side of h over the hypotenuse of 7, and that comes out to be 7 times the sine of 21 degrees.
So let's just write that out. Seven times the sine of 21 degrees. This comes out to
be approximately 11.3 square units. You might have done this work in Geometry. If you have
not, well there you go. But there is another formula that is in your law of sines section
that looks like a new formula, but indeed it is not. It says this. The area of a triangle,
it is like another version of area, the area of a triangle is one half a b times the sine
of c. And then it also says that it is equal to one have a c times the sine of B, and oh
yeah by the way it is also equal to one half b c times the sine of A. Hello, they are all
the same formula. It would be really helpful sometimes if we would get off everything has
to be a variable kick. Just write some of this stuff in words. What do we really have
here? Well first of all, before I validate to you that this old stuff that we did to
find area is exactly what this says already, before we validate that or point it out to
you if you don't see that yet... All three of these versions, all they say is that the
area of a triangle is equal to one half, these are lower case letters so they are two sides,
so how about side sub one times side sub two just to indicate there is two sides and this
is a, b, and angle c, this is a, c, and angle b, this is b, c, and angle A. Well this example
is going... I can use this formula for this example, otherwise why would I be talking
about it, the two sides are coming together to make the angle of 21 degrees. What do you
call that when you have two sides coming together to make and angle? It is an included angle.
Some vocabulary word that you have not said or heard since Geometry. But, this is two
sides of an oblique triangle. Remember we are finding the area of this large oblique
triangle, not the green temporary triangle I made to find the height. Two sides and an
included angle. So if you have two sides and an included angle of that oblique triangle,
then you can find the area by just doing one half of the two sides times the sine of the
included angle. Clearly you are not going to see this in a textbook. Your textbook is
going to write three different versions. But, they are all the same thing. Now look at what
we have got here. One half of nine, the base if you will, times seven, that is the other
side that we have, times the sine of 21 degrees. These are our two sides and this is our included
angle. So by using some skills that we knew from Geometry much less the trig that we learned,
we can see the tie in with what we did years ago in Geometry with these new formulas that
our prelacy book is giving us. We are just seeing that it is really not new, it just
looks different and thus it might be giving you something new to memorize but it is not
new. You have already been doing these formulas, only you did them in two pieces. One half
base times height and you old SOHCAHTOA stuff finding the missing sides of a right triangle.
I am Mr. Tarrou BAM!!! Go Do Your Homework. Thank you very much for watching:)