Pre-Calculus: Trig to Find Right Triangle sides

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The Trigonometric Functions
Right Angle Trigonometry
Using Trigonometric Functions to Find Unknown Sides of Right Triangles Page [1 of 1]
You know, in the real world, when you’re actually trying to compute these sines and cosines and so forth and
measuring things, triangles don’t have sort of perfect beautiful angles that work out so nicely. But, that’s okay. Years
ago, people used to use tables to look up, in fact, the sine of all sorts of weird angles. But, now that’s completely
unnecessary with the advent of just, you know, a hand-held calculator, has the sine, cosine, tangent functions. They
usually don’t have the secant, cosecant and cotangent, because it’s just literally the flip of the three that we start off
with. So, you’ll rarely see them on calculators. You can always find the cosecant, by just finding first of all, for the
cosecant, you would find the sine and then just take the reciprocal of it. One over that. So, it’s not a big deal. Now,
but to try to show how you sort of use a calculator if you wanted to. Let’s suppose you’re given the following situation.
I tell you I have a triangle. This side is 20. It’s a right triangle. And this measure is 32°. I’m using degrees here now.
And I want to know this length here. What’s the hypotenuse? Could I find that? Well, you see, so trig actually
empowers us to actually be able to solve this. So, if I just know, this side and just the measure of this angle, I can
actually now compute a length. That’s the power of trigonometry and the trigonometric functions. It allows me to find
a length of a side of a right angle if I just know one side and one angle. Remember, if I knew the length of two sides, I
can always find this using Pythagorean Theorem. But if I know the length of only one side, but I know the measure of
an angle, I should be able to find that length using trigonometry. How would I do that?
I’d say, is there some kind of a trigonometric relationship that links this angle with this opposite side and this
hypotenuse? Well, sure – sine. So, in fact, the sine of that angle, which is 32° must equal what? It must equal
opposite over hypotenuse, so
20
x
. Now that’s an equation in x that I can solve. What can you do? You can multiply
everything through by x . And then you can divide everything through by sine of 32° and I see x = 20 divided by sine
32°. A little point of caution here. Sometimes, you get so involved in pushing symbols all over, you think it’s sort of
sine multiplied by 32°. Remember, sine is a function, it’s sine of 32°. It’s like saying square root of 25 equals square
root times 25. It’s square root of 25. Think of this the same way. This is sine of 32°. Sine by itself, naked sine
doesn’t make any sense. It’s like sine of – and you’re waiting, sine of what? So you have to remember that this is not
multiplication, it’s telling you what you’re taking the sine of.
Anyway, this is now just a number and you can compute that, using a calculator. So you go on your calculator. You
make sure you’re in degrees form and then compute the sine of that and if you compute the sine of 32°, you’ll see
.529919. So take 20 divided by that and you see 37.74 (stuff). So that length is 37.74 (something.) By the way, it
should be bigger than this because the hypotenuse should always be pretty hefty compared to these guys. So that
looks pretty good.
Let’s try one more. The calculator actually comes in handy for actually looking these things up. Suppose that I tell
you I have a right triangle. This angle was measured to be 42° and I know this hypotenuse is 8, but I want to find this
length right here. I’ll call it “ l ”. How can I do that? Well, I need a relationship that links this adjacent side and the
hypotenuse to this 42°. Well, cosine sure comes in handy because the cosine of 42° equals adjacent l over
hypotenuse. So, what does l equal. I just multiply through by. Eight times cosine of 42°. And again, 42° is just
some number, so I can actually look it up or compute it on a calculator. So, you just turn on the calculator and you
type the cosine of 42° and what you get is 8 x .74314, and if you multiply that out, you get 5.945. So, this length is
5.945. So, it’s really cool. So, we can see already an application of all this trig stuff is that, if you have a right triangle
and you just know the length of one side, and an angle, you can actually find the length of any side that you want.
Because, of course, now that I have this side, I can use Pythagorean Theorem to find this. I know this angle, because
it’s going to be 90° minus the 42. So, once you know a side of a right triangle, and any angle outside of the 90° one,
you can use trig to find everything else. You can see the power of trig.
Is this really useful, interesting or important? Yes, in fact, this is used all the time and up next I’ll show you a real
world example where, in fact, you can use trig to find out these things and how they occur in the real world. I’ll see
you in the real world, up next.