Unit Circle Definition of Trig Functions Khan Academy English English (auto-generated)

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Thai sacklnwza007

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Hello. Well, welcome to the next presentation in the

hello well welcome to the I guess the next presentation and the trigonometry

trigonometry modules.

of modules just to start off a little

Just to start off a little bit, let's review what

bit let's review what we've done so far

we've done so far.

in the last couple modules we learned

In the last couple modules, we learned the definitions-- or at

the definitions or at least I guess we

least, I guess, we could call it a partial definition-- of

could call it a partial definition of

the sine, the cosine, and the tangent functions.

the sine the cosine and tangent

And the mnemonic we used to memorize that was sohcahtoa.

functions and the pneumonic we used to memorize that was sohcahtoa let me write

Let me write that down.

so Toa and what that told us is let's

And what that told us is, let's say we had a right triangle.

say we had a right triangle let me draw

Let me draw a right triangle.

a right triangle whoops

right triangle this is a right angle

This is a right angle here.

here this is the hypotenuse so let me

This is the hypotenuse. Let me label the hypotenuse, h.

just write let me label the hypotenuse H

Let me label this-- and so we want to figure out, we want to

let me label this and let's put this and say we want to figure out we want to use

use this angle right here.

this angle right here theta we'll call

Theta, we'll call this theta.

this theta whatever then this is the

Whatever. Then this is the adjacent side, and this is the opposite side.

adjacent side and this is the opposite

And that's an o.

side and this that's a No and so so

So soh tells us that sine is equal to opposite

tells us that sine is equal to opposite

over hypotenuse.

over hypotenuse cosine is equal to

Cosine is equal to adjacent over hypotenuse.

adjacent over hypotenuse and tangent is

And tangent is equal to opposite over adjacent.

equal to opposite over adjacent now that

And I think, by this point-- and especially if you did some

I think by this point especially if you

of the exercises on the Khan Academy-- that should be second

did some of the exercises on the Khan Academy that should be second nature and

nature and should make a lot of sense to you.

should make a lot of sense to you but

But this definition, using a right triangle like this,

this definition using a right triangle

actually breaks down at certain points.

like this actually break down at certain

Actually, at a lot of points.

points I can actually a lot of points

For example, what happens as this angle right here

for example what happens as this as this

approaches 90 degrees?

angle right here approaches 90 degrees

You can't have two 90 degree angles in a right

and you can't have two 90-degree angles

triangle, can you?

in a right triangle can you then it

Then it would be like a rectangle or something.

would it would be like a rectangle or

But you could actually probably figure out what happens as

something but you could actually

it approaches 90 degrees.

possibly figure out what happens as it

But the definition, this definition, breaks

approaches 90 degrees but the definition

this definition breaks down for that

Also, what happens if this angle is negative?

also what happens if this angle is

Or what happens if this angle is more than 90 degrees?

negative or what happens if this angle is more than 90 degrees or what happens

Or what happens if it's 800 degrees?

if it's 800 degrees or you know eight pi

Or you know, 8 pi radians?

radians not that 808 PI radians are the

Not that 800 and 8 pi radians are the same thing. But obviously, this definition starts to break down.

same thing but obviously this definition

Because we couldn't even draw a right triangle that

starts to break down because we couldn't

has those properties.

even draw a right triangle that has has

So now I'm going to introduce you to an extension

those properties so now I'm going to introduce you to a an extension of this

of this definition.

definition it's really the same thing

It's really the same thing, but it allows the sine, the cosine,

and the tangent functions to be defined for angles greater than

the sine the cosine and the tangent functions to be defined for angles

or equal to pi over 2, or 90 degrees, or less than 0.

greater than or equal to PI over 2 or 90 degrees or less than zero so let's draw

So let's draw a unit circle.

a unit circle so this is just the

So this is just the coordinate axis, and here is a

coordinate axis and here is a circle of

circle of radius 1. And let's make-- let me see.

radius one and let's make as we see let

Let me make sure I'm using the correct pen tool.

me make sure I'm using the correct pen

OK. So let's call this right here-- so this is theta.

tool okay so let's call this right here so this is Theta this is an angle right

This is an angle, right?

between this line between kind of the

Between the x-axis and this line I just drew here.

x-axis and this line I just drew here

And this is a radius, right?

and this is a radius right and we said

And we said that this has a radius 1.

that this is this has a radius one so

So the length of this line is 1, right?

this the length of this line is 1 right

Because it just goes from the origin to the

because it just goes from the origin to

outside of the circle.

the outside of the circle so it has a

So it has a radius of 1. And now I'm going to draw a right triangle again.

radius of 1 and now I'm going to draw a

Let me just drop a line from here.

right triangle again so let me just drop a line from here so there I have a right

So there I have a right triangle.

triangle so if we use the old definition

So if we use the old definition we learned before.

we learned before let's just focus on

Let's just focus on sine for now. So sine is equal to opposite over hypotenuse.

sine for now so sine is equal to opposite over hypotenuse let's apply

Let's apply that to this right triangle right here.

that to this right triangle right here

This is the right angle.

this is the right angle so what's the

So what's the opposite angle of this?

opposite angle of this what's the

What's the opposite side from this angle?

opposite side from this angle right it's

I'm going to change to yellow.

it is me I'm going to change to yellow

It's this side, right?

it's this side right this is the

This is the opposite side. And what's the hypotenuse?

opposite side and what's the hypotenuse

The hypotenuse is just this radius, right?

the hypotenuse is just this radius right

And let's just say that this point, where it intersects

and let's just say that this this point

the circle-- let's call this point right here x comma y. So what's the height of this opposite side?

we're two intersects the circle let's call this point right here let's call this point X comma Y all right so this what's the what's the height of this

opposite side what's Y right because

Well, it's y, right? Because it's just the height of that point.

it's just it's just the height of that

This is of height y.

point this is of height Y so sine of

So sine of this angle right here, sine of theta, is

this angle right here sine of theta is

going to equal the opposite side-- which is this yellow

going to equal the opposite side which

side, which is just the y-coordinate-- is going to

is this yellow side which is just the y-coordinate is going to equal Y over

equal y over the hypotenuse.

the hypotenuse the hypotenuse is this

The hypotenuse is this pink side here.

pink side here and what's the length of

And what's the length of the hypotenuse?

the hypotenuse what's the radius of this

Well, it's the radius of this unit circle. So it's 1.

unit circle so it's 1 and Y divided by 1

And y divided by 1? Well, that's just y.

well that's just Y so we see that sine

So we see that sine of theta is equal to y.

of theta sine of theta is equal to Y

Let's do the same thing for cosine of theta.

let's do the same thing for cosine of

Well, we know that cosine is equal to adjacent over hypotenuse.

theta well we know that cosine cosine is equal to adjacent over hypotenuse well

Well, what's the adjacent side here?

what's the adjacent side here let me I'm

I'm running out of colors.

running out of colors right the adjacent

The adjacent side is this bottom side, right here.

side is this bottom side right here so

So that would equal-- so if I said-- I'm running

that would equal so if I said I'm

out of space, too. Cosine of theta would equal this gray side-- which is

running out of space to cosine cosine of theta would equal this gray side which

the adjacent side-- and what is that?

is the adjacent side and what is that

What is this length?

what is this what what is this length

What is the length of this side?

what is the length of the side well it's

Well, it's just x, right?

just X right if this is the point XY

If this is the point x, y then this distance here is x and we

then this distance here is X and we

already learned this distance-- or we already observed--

already learned this distance or we

that this distance is y.

already observe that this distance is y

So this distance being just x, we know that the

so this distance being just X we know

adjacent length is x.

that the adjacent length is X so we say

So we say cosine of theta is equal to x over the hypotenuse.

cosine theta is equal to x over the

And once again, the hypotenuse is 1.

hypotenuse and once again the hypotenuse

So cosine of theta is equal to x.

is 1 so cosine of theta is equal to X I

I know what you're thinking.

know what you're thinking Sal that's

Sal, that's very nice and cute.

that's very nice and cute cosine of

Cosine of theta equals x, sine of theta equals y.

theta equals x sine theta equals y Y but

But how is this really different from what we

how is this really different from what

were doing before?

we were doing before well if I define it

Well, if I define it this way, now all of a sudden when the

this way now all of a sudden

angle becomes 90 degrees, now I can actually define

when the angle becomes let's say 90 degrees let's say this is now I can

what sine of theta is.

actually define what sine of theta is

Sine of theta now is just y.

sine of theta now is just Y is just the

Is just the y-coordinate, which is 1.

y-coordinate which is 1 if theta is

If theta is equal to-- I'm going to make sure it's very

equal to I'm going to make sure it's

messy right here-- if theta is equal to 90 degrees,

very messy right here if theta is equal

or pi over 2 radians.

to 90 degrees or PI over 2 radians all

This is pi over 2.

right this is PI over 2 this angle right

This angle right here. And similar, cosine of pi over 2 is 0.

here right and similar cosine of PI over

Because the x-coordinate right here is 0.

2 is 0 because the x coordinate right here is 0 let me do it with a couple

Let me do it with a couple more examples.

more examples I know I'm forgetting the

Oh, I'm forgetting the tangent function.

tangent function and you could probably

And you could probably figure out now, what is the

figure out now what is the definition

definition now we can use for the tangent function?

now we could use for the tangent

Well, going back-- let's use this green theta here.

function well going back let's use this green

Because it's kind of a normal angle.

theta here because it's kind of a normal

So in this green angle here, tangent is

angle so this green theta here tangent

opposite over adjacent.

is opposite over adjacent so tangent now

So tangent now, we can define as y/x.

we can define as Y over X and remember

And remember, these y's and x's that we're using are the point

these y's and x's that we're using are

on the unit circle where the angle that's defined by this--

the point on the unit circle where the angle that's defined by this by whatever

by whatever-- where the radius that is subtended by this

it where the the radius that is

angle, or I guess the arc, intersects-- actually,

subtended by this angle or I guess the arc intersects actually I'm getting

I'm getting confused with terminology.

confuse with terminology it's where this

It's where this line intersects the circumference.

line intersects the circumference the

The coordinate of that-- the sine of theta is equal to y.

coordinate of that site the sine of theta is equal to Y the cosine of theta

The cosine of theta is equal to x.

is equal to X and the tangent of theta

And the tangent of theta is equal to y/x.

is equal to Y over X let's do a couple

Let's do a couple of examples and hopefully this'll make a

of examples and and hopefully this will

little bit more sense to you.

make a little bit more sense to you let

Let me try to really fast draw a new unit circle.

me try to really fast draw a new unit

So that's my unit circle. And here's the coordinate axis.

circle so let's so that's my unit circle and here's the coordinate axis was one

It's one of them. And here is the other one.

of them and here is the other one so let's get an example like so if we use

So if we use the angle-- let's use the angle pi over 2, right?

the angle let's use the angle PI over 2

Theta equals pi over 2.

right theta equals PI over 2 well PI over 2 is

Well, pi over 2 is right here.

It's a 90 degree angle, if you wanted to use degrees.

right this it's a 90 degree angle if you wanted to use degrees and now we just

And now, we just figure out where it intersects

figure out where it intersects the unit

the unit circle.

circle I want to again this is a unit

And once again, this is a unit circle, so it has a radius of 1.

circle so as radius is 1 so we can see

So we can see that sine of pi over 2 equals the y-coordinate

that sine of PI over 2 it equals the

where it intersects the unit circle. So that's just 1.

y-coordinate where it intersects the unit circle so that's just 1 what's

What's cosine of pi over 2?

cosine of PI over 2 well it's just the

Well, it's just the x-coordinate, where you

x-coordinate where you intersect the

intersect the unit circle.

unit circle and the x-coordinate here is

And the x-coordinate here is 0. And what's the tangent of pi over 2?

0 and what's the tangent of PI over 2

This is interesting.

this is interesting

The tangent of pi over 2.

the tangent of PI over 2 well the

Well, the tangent we defined now as y/x.

tangent we define now as Y over X so the

So the y-coordinate, this is the point 0, 1, right?

y-coordinate this is the point 0 1 right

The y-coordinate is 1.

the y-coordinate is 1 so it equals 1

So it equals 1/0. So this is undefined.

over 0 yep so this is undefined so still

So still, we don't have a tangent function that can

we don't have a tangent function that

define itself at certain points.

can define itself at certain points but

But in the next module, we're actually going to graph this.

I've in the next module we're actually

And you'll see that it approaches infinity.

going to graph this and you'll see that it it approaches infinity and similarly

And similarly, we could try to find the functions for when

we could try to find the functions for

theta equals pi, right?

when theta equals pi right that's like

That's like 180 degrees.

180 degrees that's this point right here

That's this point right here. So sine of pi.

What's the y-coordinate at this point?

what's the y-coordinate at this point

Well, this point is negative 1 comma 0.

well this point is negative 1 comma 0 so

So the y-coordinate is 0.

the y-coordinate is 0

What's the x-coordinate? Cosine of pi.

what's the x-coordinate cosine of pi

That's negative 1.

that's negative 1 and of course what's

And of course, what's the tangent of pi radians?

the tangent of pi radians it's Y over X

It's y/x. So it's 0 over negative 1, which equals 0.

so it's 0 over negative 1 equals 0

Hopefully this makes sense.

hopefully this makes sense now in the

Now in the next module I'll actually graph these points.

next module I'll actually graph these

And you'll see how it all comes together and why it is useful

points and you'll see I guess how it all comes together and and why it is useful

to define the sine, the cosine, the tangent functions this way.

to define the sine the cosine and tangent functions this way see you soon

See you soon. Bye.

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