Basic Trigonometry II

Another example of figuring out the sine, cosine, and tangent of an angle in a right triangle

Welcome to the second part
of the presentation on

basic trigonometry.

In the last 10 minutes, I had
trouble getting in a lot of

examples, so I want to do a
couple more with you guys.

OK.

So, let me start over just
because this got messy.

And we're just going to do what
we did in the last time around.

So let me just draw
another right triangle.

And make sure it's not
going to be too big.

Here's my right triangle.

And let me just throw
out some random sides.

Let me say that this is 6.

Let's make this side 5.

And then, if this is a right
triangle, Pythagorean Theorem

tells us that this would be
the square root of what?

36 plus 25 is equal to
the square root of 61.

I think that's right.

I've gotten feedback on some of
my other videos that I tend to

get this type of
addition wrong.

I malfunction sometimes.

But anyway.

So this side is the square
root of 61, and that's

the hypotenuse.

So let's get started
with some problems.

If I were to give you--
if I were to ask you.

Well, let's see.

Let's call this angle theta.

And I want to know what
is the tangent of theta?

And we'll shorten that as
tangent of-- tan of theta.

What is the tangent of
this angle right here?

Well, you probably already
forgot what the definition

of tangent is.

So I will repeat it.

In fact, I will write
up in this corner.

Soh cah toa.

So I think now your brain might
be refreshed and you'll

remember that toa is the
mnemonic for tangent.

And it says that tangent
is equal to the opposite

over the adjacent.

So the tangent of theta is
equal to the opposite side--

well, that's this side, that's
the side of length 5--

over the adjacent side.

That's this side.

The side of length 6.

That's pretty easy, huh?

The tangent of theta is 5/6.

And we'll just do
a couple more.

All right?

We'll just go through all of
the trig functions, or at least

the basic trig functions.

What is the sine of theta?

Well, let's go back
to our mnemonic.

Soh cah toa.

This is one of the few things
in mathematics that you

should probably memorize.

It's kind of a
funny word anyway.

And soh-- to find sine.

It tells us that sine is
opposite over hypotenuse.

Well, the opposite side,
once again, is 5.

And what's the hypotenuse?

Well, the hypotenuse, we
just figured out, was

the square root of 61.

And a lot of people don't like
irrational denominators.

So we can rationalize
the denominator.

And we do that by multiplying
the numerator and the

denominator by the
square root of 61.

So if we say that this is equal
to 5 over the square root of 61

times the square root of
61, over the square

root of 61, right?

We're just multiplying it by 1.

Because this is the same
thing top and bottom.

This equals 5 square roots of
61 over-- what's the square

root of 61 times the
square root of 61?

Oh yeah, it's 61.

So the sine of theta is 5
square roots of 61 over 61.

And then finally, let me
make some space here.

Let me erase some stuff.

Let me erase this
one right here.

And you're probably still
wondering, OK, I kind

of get this whole sine,
tangent, cosine thing.

What is it useful for?

And all I can tell you right
now is, get to know how to use

these, soh cah toa, and in the
next presentation and onwards,

we're going to show you that
trigonometry is actually

probably one of the most
obviously useful

things in math.

You can figure out
all sorts of things.

How far planets are, how
tall buildings are.

I mean, there's tons of
things you could figure

out with trigonometry.

And then later, we'll study
sine waves and cosine

waves, and all that.

You'll learn that it actually
describes almost everything.

But anyway.

Going back to the problem.

All we have left now is cosine.

Oh, look how big that is.

Cosine of theta equals-- we'll
go back to our mnemonic.

Soh cah toa.

Well, cosine is adjacent
over hypotenuse.

So once again, what's
the adjacent side?

Well, this is the angle we're
finding the cosine of, so the

adjacent side is right here.

So length 6.

So it equals the adjacent
side, which is 6, right?

And we figured out what
the hypotenuse was.

That's this side.

And its length,
square root of 61.

And if we rationalize this
denominator, we get 6 square

roots of 61 over 61.

It's kind of messy numbers.

But I think now you get the
hang of figuring out-- if you

know the sides of a triangle--
figuring out what the sine, the

cosine, or the tangent of any
given angle in that

right triangle is.

And obviously, you can't figure
it out for this angle, because

for this angle the opposite
and the hypotenuse are

actually the same number.

So actually-- never mind.

You actually can figure it out.

But it actually gives
something-- an

interesting number.

So with that said, I will
finish this presentation.

And in the next presentation, I
will show you how-- if we know

what the sine, or the cosine,
or the tangent of an angle is,

and we know one of the sides--
how we can figure out

the other sides.

See you in the next
presentation.

Bye.