Pre-Calculus: The Law of Sines

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Applications of Trigonometry
The Law of Sines
The Law of Sines Page [1 of 2]
A lot of the trig functions we develop—sine, cosine, tangent, cosecant, secant, cotangent—all those functions in fact
all arose from looking at right triangles, and looking at ratios of sides of right triangles and hypoteni of right triangles,
and seeing how those compare. So what if you have a triangle that’s not a right triangle? Not all triangles are right.
Some triangles are wrong! Now, what can you say there? Can you say any kind of trig-type stuff there? Well, the
answer is yes.
The first thing I want to tell you about is known as the Law of Sines. What is the Law of Sines? This allows us to
consider sines of angles of triangles that in fact are not right triangles. So suppose we have a triangle that looks like
this. I’m going to try to make this deliberately not a right triangle. Does that look right to you? I hope not. That
should be a really big angle. This is not a right triangle, not a right triangle.
Let me give some labels to these angles here. Let me call this angle ? , the Greek letter Alpha, and the opposite side
I’m going to call that length a. So I’m trying to get into a habit of using a Greek letter for an angle and then the
opposite side is related in regular English or Arabic. All right. So if I call this ? , for example—if I call this angle Beta,
then I call this little b. And if I call this angle ? (Gamma), I’d call this little c. So always it’s important to understand
that I’m always going to think of my triangle where if I say what Alpha equals, Alpha is some angle and then I tell you
the length of a, that means the opposite side of this angle.
Well, what the Law of Sines says is the relationship between the sine of all these angles and the sides, and here’s
what the Law of Sines says. The Law of Sines says that if you take sin? , take sine of this angle, and divide it by the
length of a, that will be equal to the sine of Beta divided by the length of b, and that in turn will be equal to the sine of
Gamma divided by the length of c. This is the Law of Sines, and it holds for any triangle. It doesn’t even have to be a
right triangle. It actually empowers us to figure out all the measurements of a triangle just given a few. Let me try to
illustrate that now with an example.
So suppose we know that a, that side, is equal to a length of 10, and ? , the angle opposite that side, is equal to 40°.
And suppose that ? we are told is 46°. So I just know this angle, that angle, and this side. That should determine
this triangle precisely, and the question is, how can we figure out this angle and the lengths of these two sides?
Well, the Law of Sines is going to be able to help us. First of all, we can figure out that third angle pretty easily
because I remember that the sum of all the angles of a triangle in fact is 180°. In fact, do you know about that?
Maybe I should just take a second to say a few words about that? You think I should? So hold on a second. Stay
right there. Don’t go away.
If I just take a piece of paper here—this is so much fun, I’ve got to do this. Take a random triangle out of a piece of
paper. How about this one here? Here’s a random triangle. I’m cutting this out live. I’m trying not to make it a right
triangle. Okay, there’s a random triangle. If you look at all the angles, let’s see what all the angles add up to. You
can just cut them out. Just cut out the angle. Whoop, there’s one angle right there. Cut this angle out; here’s another
angle. And then cut this angle out, whoop. You see those are all the angles of the triangle. Now, if you just add up
all the angles, let’s see what happens, okay? So I’m just going to add up all those angles. So I take this angle plus
this angle, and when I add this angle, look what happens. Heh, heh, heh. Isn’t that cool? It’s 180° always. Always.
So I just proved that to you. That’s sort of fun.
Anyway, we know that if we know two angles, that gives us the third angle for sure. I just add these two things up and
get 86, and then take 180° and subtract 86. When I do that, I would see this angle, which in this case would be 94.
So I immediately can figure out that ? = 94 ° because the sum has to be 180. So the angles are pretty easy to find.
Well, what about the length? How could I find the length of b and the length of c? Well, I’m going to use the Law of
Sines. Now, what do I know? I know that this is 40° and this length is 10. So that means I can insert that information
here and say that
sin 400
10
will equal—and now I know what ? is. That’s sin 46° divided by that length, b. And look,
now I have an equation that everything is a number except for b. So using a calculator, I can actually solve this. How
Applications of Trigonometry
The Law of Sines
The Law of Sines Page [2 of 2]
would I solve this? Well, I would see that if I multiply through by b and divide a little bit, I would see that—multiply
through by b. I’d get a b here. If I divide I’d see 0
sin 46
sin 40
o
and I can multiply all that by 10. So if you solve this for b
carefully—so invert that and bring it here, and bring the b up here—you’d see that. And now I just calculate that on
the calculator. Make sure you’re set on degrees and figure out sin 46, divide that by sin 40, and multiply all that by
10, and you get your answer. And what is it? I seem to be getting a slightly different—actually I computed this in
advance, you know? I’m getting a slightly different answer here so I’m just making sure that everything is completely
happy here. Oh, yes, I see. Everything is fine. I get b equals 11.1909.
So a was 10-something, b is 11-something. By the way, that’s consistent. Because you know what? If the angle is
big, then it makes sense that that other side should be big, right? Because that angle is spread out more; it needs a
lot more room. So the bigger the angle, the bigger the side. This is a 40° angle that’s facing this 10-unit side, so a
46° angle should be facing a really bigger side, so it’s 11-something.
And now I can actually find out the c. How would I find out the c? Well, I’d do the same thing. I know that
sin 400
10
is
going to equal the sine of—now it’s 94° divided by c. So I can do the exact same procedure now, and what I see is
sin 40
10
o
, that’s this thing, putting in this information—equals the sine of that third angle which I computed to be 94°,
divided by c. So what does c have to equal? C has to equal 10sin94°. I’m just now solving this for c. You can do
that any way you want. I’m doing it pretty quickly. And if I do that, let’s see what we get. Whoops, I am in graphing
mode, which is so sad because I want to be in regular mode, and I have no idea—did you ever get into graphing
mode and you just can’t get out, so your whole life is sort of graphing mode for a while?
All right, so it would be 10sin94° divided by—make sure, again, you’re set on degrees and not radians. And I see
that c would equal 15.51-something. And that’s even bigger. But notice that’s consistent because, of course, that’s a
huge angle. This is a really big angle so this should be a really long side. It’s 15.51. b, the middle length side, is
11.1-something, and a, of course, is given to be 10. So just knowing a and these two angles, I was able to use the
Law of Sines to figure out the others, and also use the fact that all the angles of a triangle add to 180 to find out the
third angle.
So the Law of Sines actually allows us to look at triangles now that are no longer right, and actually start to prove
properties about them and compute their lengths and angles. We’ll take a look at more of these things up next.