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.
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 that you're going to find the trig function of.
Let's say you wanted to find the sine and s-i-n
is short for sine.
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.
Now, that's going to be confusing right now, but
we're going to do a lot of examples, and I think
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?
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.
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.
Well, once again, let's figure out what the adjacent side is.
Well, the adjacent side-- this side was the opposite side,
right, because it's opposite the angle.
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.
OK, one left.
The TOA.
So if you remember what I said a little while ago-- well,
we'll figure it out.
But what is the-- oh, look how big that is.
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!