Basic Trigonometry

An introduction to trigonometric functions: sine, cosine, and tangent.

Welcome to the presentation
on basic trigonometry.

Sorry it's taken so long to get
a new video out, but I had

a lot of family in town.

So let's get started
with trigonometry.

Let me get the pen
tools all set up.

I'm a little rusty.

I'll use green.

Trigonometry.

I think it means-- I think it's
from Ancient Greek, and it

means triangle measure.

I think that I read it on
Wikipedia a couple days ago,

so I believe that's the case.

But all trigonometry is is
really the study of right

triangles and the relationship
between the sides and the

angles of a right triangle.

That might sound a
little confusing, but

I'll get started.

And actually, you've probably
seen a lot of these things that

we're going to go over now, and
you'll finally know what

those buttons on the
calculator actually do.

So let's start with
a right triangle.

Let's see, so it's a triangle.

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00:01:01,62 --> 00:01:05,42
And it's a right triangle.

Just for simplicity, let's say
that this side is 3, this side

is 4, and the hypotenuse is 5.

So the trig functions tell you
that for any angle, it tells

you what the ratios of the
sides of the triangle are

relative to that angle.

Let me give you an example.

Let's call this angle theta.

Theta is the Greek alphabet
people tend to use for the

angle you want to take the trig function off.

Let's say you wanted to find the sine of theta. Sin is a short form.

Let's say you wanted to
find the sine of theta.

So before we solve the sine
of theta, I'm just going to

throw out a mnemonic that I
remembered when I was learning

this in school, and I carried
it every time, and every time I

do a trig problem, I actually
write it down on the page, or I

at least repeat it to myself.

And this is SOH CAH TOA.

I have vague memories of my
math teacher in high school

telling a story about some
Indian princess, who was named

Sohcahtoa, but I forget.

But all you have to
remember is SOHCAHTOA.

Now you might say, well,
what it's SOHCAHTOA?

Well, SOHCAHTOA says that sine
is opposite over hypotenuse,

cosine is adjacent over
hypotenuse, and tangent is

opposite over adjacent.

it's going to make sense.

So let's go back
to this problem.

We want to know what's
the sine of theta.

Theta is this angle
in the triangle.

So let's go to our
mnemonic SOHCAHTOA.

So which one is sine?

Well, S for sine,
so we use SOH.

And we know that sine from
this mnemonic, sine of, let's

say, theta, is equal to
opposite over hypotenuse.

Opposite over hypotenuse.

OK, so let's just figure
out what the opposite

and the hypotenuse are.

Well, what is the opposite
side of this angle?

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00:03:14,45 --> 00:03:17,5
Well, if we just go opposite
the angle, let's go here,

the opposite side is 4,
is this length of 4.

I'll make that in a color.

Oh, I thought I was
changing colors.

Yeah, so this side is the
opposite, and I'll circle it.

Now, which side is
the hypotenuse?

And you know this one.

We've been doing this in the
Pythagorean theorem modules.

The long side, or the side
opposite the right angle,

is the hypotenuse.

So that is the hypotenuse.

So now I think we're
ready to figure out what

the sine of theta is.

The sine-- whoops,
I stayed in pink.

Sine of theta is equal to the
opposite side, 4, over the

hypotenuse, which is 5.

We're done.

Let's figure out what-- let me
erase part of this, and we'll

figure out some more things
about this triangle.

Let me erase this.

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00:04:14,8 --> 00:04:17,21
I think if you practice this,
you'll realize that this is

probably one of the easier
things you learn in

mathematics, and it's actually
shocking that they take-- that

they wait until Precalculus to
teach this, because a smart

middle-schooler could, I think,
easily handle this.

Not to make you feel bad if
you're not getting it, just to

give you confidence that you
will get it, and you'll realize

that it is very simple.

OK, so let's figure out
what the cosine-- and

cos is short for cosine.

I'll write it out, but I'm
sure you've seen it before.

So what is the cosine of theta?

Well, we go back to our
mnemonic: SOHCAHTOA.

Well, cosine is the CAH, right?

And that tells us that
cosine of theta is equal to

adjacent over hypotenuse.

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00:05:11,73 --> 00:05:14,78
Well, once again, let's figure
out what the adjacent side is.

This side is the hypotenuse,
because it's the longest side,

and then, you could, just by
deductive reasoning, but also

just by looking at it, you see
that this side right here, the

side of length 3, is adjacent
to the angle, right?

Adjacent means right beside it.

So that's the adjacent side.

We already figured out that
the hypotenuse is that

side that I wrote in pink.

So we're ready to figure out
what cosine of theta equals.

Cosine of theta is equal to the
adjacent side, that's 3, over

the hypotenuse, which
is this pink side, 5.

Pretty straightforward,
isn't it?

Let's do another one.

OK, I don't want to
erase the whole thing.

I just want to erase part of
the page because I want to

keep using this triangle.

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00:06:18,31 --> 00:06:19,92
OK, one left.

The TOA.

So if you remember what I said
a little while ago-- well,

What is the tan of theta,
or the tangent of theta?

Well, let's go back
to our mnemonic.

TOA, right?

TOA is for tangent,
or t for tangent.

So it tells us that tangent is
the opposite over the adjacent.

So tan of theta is equal to
opposite over adjacent.

Well, that equals-- what
was the opposite side?

Right, the opposite side was 4.

And what was the adjacent side?

Well, we just saw that.

It was 3.

So the tangent of
this angle is 4/3.

Now let's do another
angle on this.

Let's call this angle
here-- I don't know.

Let's call it x.

I don't know any
other Greek letters.

Let's call that angle x.

So if we want to figure out the
tan of x, let's see if it's

the same as the tan of theta.

The tan of x.

Well, now what's
the opposite side?

Well, now the opposite side
is the white side, right?

Because opposite this
angle is the 3 side.

So we see here tan is opposite
over adjacent, so opposite is

3, and then adjacent is 4.

This is interesting.

The tangent of this angle
is the inverse of the

tangent to that angle.

I don't want to confuse you too
much, but I just want to show

you that when you take the trig
functions, it matters which

angle of the right angle you're
taking the functions of.

And you might be saying, well,
this is all good and well,

Sal, but what use is this?

Well, we'll later show you that
if you have some of the

information, so you know an
angle, and you know a side, or

you know a couple of sides, you
can figure out-- and if you

have either a slide ruler or a
trig table or a good

calculator, you can figure
out-- given the sides of a

triangle, you can figure out
the angles, or given an angle

and a side, you could
figure out other sides.

And we're actually going to
do that in the next module.

So, hopefully, this gives you a
little bit of an introduction.

I'm almost out of time on the
YouTube 10-minute limit, so I'm

going to wait to do a couple
more examples in

the next video.

See you in the next
presentation.

Bye!