Welcome to the presentation on basic trigonometry.
Welcome to the presentation on basic trigonometry.

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

a lot of family in town.
a lot of family in town.

So let's get started with trigonometry.
So let's get started with trigonometry.

Let me get the pen tools all set up.
Let me get the pen tools all set up.

I'm a little rusty.
I'm a little rusty.

I'll use green.
I'll use green.

Trigonometry.
Trigonometry.

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

means triangle measure.
means triangle measure.

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

so I believe that's the case.
so I believe that's the case.

But all trigonometry is is really the study of right
But all trigonometry is is really the study of right

triangles and the relationship between the sides and the
triangles and the relationship between the sides and the

angles of a right triangle.
angles of a right triangle.

That might sound a little confusing, but
That might sound a little confusing, but

I'll get started.
I'll get started.

And actually, you've probably seen a lot of these things that
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
we're going to go over now, and you'll finally know what

those buttons on the calculator actually do.
those buttons on the calculator actually do.

So let's start with a right triangle.
So let's start with a right triangle.

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

And it's a right triangle.
And it's a right triangle.

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

is 4, and the hypotenuse is 5.
is 4, and the hypotenuse is 5.

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

you what the ratios of the sides of the triangle are
you what the ratios of the sides of the triangle are

relative to that angle.
relative to that angle.

Let me give you an example.
Let me give you an example.

Let's call this angle theta.
Let's call this angle theta.

Theta is the Greek alphabet people tend to use for the
Theta is the Greek alphabet people tend to use for the

angle that you're going to find the trig function of.
angle that you're going to find the trig function of.

Let's say you wanted to find the sine and s-i-n
Let's say you wanted to find the sine and s-i-n

is short for sine.
is short for sine.

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

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

throw out a mnemonic that I remembered when I was learning
throw out a mnemonic that I remembered when I was learning

this in school, and I carried it every time, and every time I
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
do a trig problem, I actually write it down on the page, or I

at least repeat it to myself.
at least repeat it to myself.

And this is SOH CAH TOA.
And this is SOH CAH TOA.

I have vague memories of my math teacher in high school
I have vague memories of my math teacher in high school

telling a story about some Indian princess, who was named
telling a story about some Indian princess, who was named

Sohcahtoa, but I forget.
Sohcahtoa, but I forget.

But all you have to remember is SOHCAHTOA.
But all you have to remember is SOHCAHTOA.

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

Well, SOHCAHTOA says that sine is opposite over hypotenuse,
Well, SOHCAHTOA says that sine is opposite over hypotenuse,

cosine is adjacent over hypotenuse, and tangent is
cosine is adjacent over hypotenuse, and tangent is

opposite over adjacent.
opposite over adjacent.

Now, that's going to be confusing right now, but
Now, that's going to be confusing right now, but

we're going to do a lot of examples, and I think
we're going to do a lot of examples, and I think

it's going to make sense.
it's going to make sense.

So let's go back to this problem.
So let's go back to this problem.

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

Theta is this angle in the triangle.
Theta is this angle in the triangle.

So let's go to our mnemonic SOHCAHTOA.
So let's go to our mnemonic SOHCAHTOA.

So which one is sine?
So which one is sine?

Well, S for sine, so we use SOH.
Well, S for sine, so we use SOH.

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

say, theta, is equal to opposite over hypotenuse.
say, theta, is equal to opposite over hypotenuse.

Opposite over hypotenuse.
Opposite over hypotenuse.

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

and the hypotenuse are.
and the hypotenuse are.

Well, what is the opposite side of this angle?
Well, what is the opposite side of this angle?

Well, if we just go opposite the angle, let's go here,
Well, if we just go opposite the angle, let's go here,

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

I'll make that in a color.
I'll make that in a color.

Oh, I thought I was changing colors.
Oh, I thought I was changing colors.

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

Now, which side is the hypotenuse?
Now, which side is the hypotenuse?

And you know this one.
And you know this one.

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

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

is the hypotenuse.
is the hypotenuse.

So that is the hypotenuse.
So that is the hypotenuse.

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

the sine of theta is.
the sine of theta is.

The sine-- whoops, I stayed in pink.
The sine-- whoops, I stayed in pink.

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

hypotenuse, which is 5.
hypotenuse, which is 5.

We're done.
We're done.

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

figure out some more things about this triangle.
figure out some more things about this triangle.

Let me erase this.
Let me erase this.

I think if you practice this, you'll realize that this is
I think if you practice this, you'll realize that this is

probably one of the easier things you learn in
probably one of the easier things you learn in

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

they wait until Precalculus to teach this, because a smart
they wait until Precalculus to teach this, because a smart

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

Not to make you feel bad if you're not getting it, just to
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
give you confidence that you will get it, and you'll realize

that it is very simple.
that it is very simple.

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

cos is short for cosine.
cos is short for cosine.

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

So what is the cosine of theta?
So what is the cosine of theta?

Well, we go back to our mnemonic: SOHCAHTOA.
Well, we go back to our mnemonic: SOHCAHTOA.

Well, cosine is the CAH, right?
Well, cosine is the CAH, right?

And that tells us that cosine of theta is equal to
And that tells us that cosine of theta is equal to

adjacent over hypotenuse.
adjacent over hypotenuse.

Well, once again, let's figure out what the adjacent side is.
Well, once again, let's figure out what the adjacent side is.

Well, the adjacent side-- this side was the opposite side,
Well, the adjacent side-- this side was the opposite side,

right, because it's opposite the angle.
right, because it's opposite the angle.

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

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

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

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

Adjacent means right beside it.
Adjacent means right beside it.

So that's the adjacent side.
So that's the adjacent side.

We already figured out that the hypotenuse is that
We already figured out that the hypotenuse is that

side that I wrote in pink.
side that I wrote in pink.

So we're ready to figure out what cosine of theta equals.
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
Cosine of theta is equal to the adjacent side, that's 3, over

the hypotenuse, which is this pink side, 5.
the hypotenuse, which is this pink side, 5.

Pretty straightforward, isn't it?
Pretty straightforward, isn't it?

Let's do another one.
Let's do another one.

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

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

keep using this triangle.
keep using this triangle.

OK, one left.
OK, one left.

The TOA.
The TOA.

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

we'll figure it out.
we'll figure it out.

But what is the-- oh, look how big that is.
But what is the-- oh, look how big that is.

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

Well, let's go back to our mnemonic.
Well, let's go back to our mnemonic.

TOA, right?
TOA, right?

TOA is for tangent, or t for tangent.
TOA is for tangent, or t for tangent.

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

So tan of theta is equal to opposite over adjacent.
So tan of theta is equal to opposite over adjacent.

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

Right, the opposite side was 4.
Right, the opposite side was 4.

And what was the adjacent side?
And what was the adjacent side?

Well, we just saw that.
Well, we just saw that.

It was 3.
It was 3.

So the tangent of this angle is 4/3.
So the tangent of this angle is 4/3.

Now let's do another angle on this.
Now let's do another angle on this.

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

Let's call it x.
Let's call it x.

I don't know any other Greek letters.
I don't know any other Greek letters.

Let's call that angle x.
Let's call that angle x.

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

The tan of x.
The tan of x.

Well, now what's the opposite side?
Well, now what's the opposite side?

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

Because opposite this angle is the 3 side.
Because opposite this angle is the 3 side.

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

3, and then adjacent is 4.
3, and then adjacent is 4.

This is interesting.
This is interesting.

The tangent of this angle is the inverse of the
The tangent of this angle is the inverse of the

tangent to that angle.
tangent to that angle.

I don't want to confuse you too much, but I just want to show
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
you that when you take the trig functions, it matters which

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

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

Sal, but what use is this?
Sal, but what use is this?

Well, we'll later show you that if you have some of the
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
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
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
have either a slide ruler or a trig table or a good

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

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

and a side, you could figure out other sides.
and a side, you could figure out other sides.

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

So, hopefully, this gives you a little bit of an introduction.
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
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
going to wait to do a couple more examples in

the next video.
the next video.

See you in the next presentation.
See you in the next presentation.

Bye!
Bye!