Basic Trigonometry II

One of the most popular Khan Academy videos, Basic Trigonometry doesn't have any subtitles - help translate it into as many languages possible!

Let's just do a ton of more examples, just so we
make sure that we're getting

this trig function thing down well.

So let's construct ourselves some right triangles.

Let's construct ourselves some right triangles, and I want to be very clear the way I've defined

it so far, this will only work in right triangles,
so if you're trying to find

the trig functions of angles that aren't part of right triangles, we're going to see that we're going to

have to construct right triangles, but let's just focus on the right triangles for now.

So let's say that I have a triangle, where
let's say this length down here is seven,

and let's say the length of this side up here, let's say that that is four.

Let's figure out what the hypotenuse over here is going to be. So we know

-let's call the hypotenuse "h"-

we know that h squared is going to be equal
to seven squared plus four squared, we know

that from of the Pythagorean theorem,

that the hypotenuse squared is equal to

the square of each of the sum of the squares

of the other two sides. h squared is equal to seven
squared plus four squared.

So this is equal to forty-nine

plus sixteen,

forty-nine plus sixteen,

forty nine plus ten is fifty-nine, plus
six is

sixty-five. It is sixty five so this h squared,

let me write: h squared

-that's different shade of yellow- so we have h squared is equal to

sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

is sixty-five, or we could say that h is equal to, if we take the square root of

both sides

square root

square root of sixty five. And we really can't simplify
this at all

this is thirteen

this is the same thing as thirteen times five,
both of those are not perfect squares and

they're both prime so you can't simplify this any more.

So this is equal to the square root

of sixty five.

Now let's find the trig, let's find the trig functions for this angle
up here. Let's call that angle up there theta.

So whenever you do it

you always want to write down - at least for
me it works out to write down -

"soh cah toa".

soh...

...soh cah toa. I have these vague memories

of my

trigonometry teacher, maybe I've read it in some
book, I don't know - you know, some, about

some type of indian princess named "soh cah toa" or whatever, but it's a very useful

mnemonic, so we can apply "soh cah toa". Let's find

let's say we want to find the cosine. We want to find the cosine of our angle.

we wanna find the cosine of our angle, you
say: "soh cah toa!"

So the "cah". "Cah" tells us what to do with cosine,

the "cah" part tells us

that cosine is adjacent over hypotenuse.

Cosine is equal to adjacent

over hypotenuse.

So let's look over here to theta; what side is adjacent?

Well we know that the hypotenuse

we know that that hypotenuse is this side over here

so it can't be that side. The only other side that's kind of adjacent to it that

isn't the hypotenuse, is this four.

So the adjacent side over here, that side is,

it's literally right next to the angle, it's one of
the sides that kind of forms the angle

it's four

over the hypotenuse.

The hypotenuse we already know is square root
of sixty-five, so it's four

over

the square root of sixty-five.

And sometimes people will want you to rationalize
the denominator which means they don't like

to have an irrational number in the denominator,
like the square root of sixty five

and if they - if you wanna rewrite this without
a

irrational number in the denominator, you can
multiply the numerator and the denominator

by the square root of sixty-five.

This clearly will not change the number, because we're multiplying it by something over itself, so we're

multiplying the number by one. That won't change
the number, but at least it gets rid of the

irrational number in the denominator. So the numerator
becomes

four times the square root of sixty-five,

and the denominator, square root of sixty five times
square root of sixty-five, is just going to be sixty-five.

We didn't get rid of the irrational number, it's still
there, but it's now in the numerator.

Now let's do the other trig functions

or at least the other core trig functions. We'll
learn in the future that there's a ton of them

but they're all derived from these

so let's think about what the sign of theta is. Once again
go to "soh cah toa"

the "soh" tells what to do with sine. Sine is opposite over hypotenuse.

Sine is equal to

opposite over hypotenuse. Sine is opposite over hypotenuse.

So for this angle what side is opposite?

We just go opposite it, what it opens into, it's opposite
the seven

so the opposite side is the seven.

This right here - that is the opposite side

and then in the

hypotenuse, it's opposite over hypotenuse. the hypotenuse is the

square root of sixty-five

and once again if we wanted to rationalize this,
we could multiply times the square root of sixty-five

over the square root of sixty-five

and the the numerator, we'll get seven square root of sixty-five
and in the denominator we will get just

sixty-five again.

Now let's do tangent!

Let us do tangent.

So if i ask you the tangent

of - the tangent of theta

once again go back to soh cah

toa the toa part tells us what to do a tangent

it tells us

it tells us that tangent

is equal to opposite over adjacent is equal
to opposite

over

opposite over adjacent

so for this angle

what is opposite we've already figured it
out it's seven it opens into the seventh opposite

the seven

so it's seven

over what side is adjacent

well this four is adjacent

this four is adjacent so the adjacent side is
four

so it's seven

over four

and we're done

we figured out all of the trig ratios for
theta let's do another one

let's do another one. i'll make it a little bit concrete
'cause right now we've been saying oh was

tangent of x, tangent of theta. let's make it a little bit more concrete

let's say

let's say, let me draw another right triangle

that's another right triangle here

everything we're dealing with

these are going to be right triangles

let's say the hypotenuse

has length four

let's say that this side over here

has length two

and let's say that this length over here is goint to be two times the square root of three we can

verify that this works

if you have this side squared so you have let
me write it down two times the square root of

three squared

plus two squared is equal to what

this is

two there's going to be four times three

four times three plus four

and this is going to be equal to twelve plus
four is equal to sixteen and sixteen is indeed

four squared so this does equal four squared

it does equal four squared it satisfies the pythagorean theorem

and if you remember some of your work from thirty
sixty ninety triangles that you might have

learned in geometry you might recognize that
this

is a thirty sixty ninety triangle this
right here is our right angle i should have

drawn it from the get go to show that this
is a right triangle

this angle right over here is our thirty degree
angle

and then this angle up here, this angle up here
is

a sixty degree angle

and it's a thirty sixteen ninety because

the side opposite the thirty degrees is half the hypotenuse

and then the side opposite the sixty degrees
is a squared three times the other side

that's not the hypotenuse

so that's that we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles

although i just did it

let's actually find the trig ratios
for the different angles

so if i were to ask you

or if anyone were to ask you what is

what is the sine of thirty degrees

and remember thirty degrees is one of the
angles in this triangle but it would apply

whenever you have a thirty degree angle and
you're dealing with the right triangle we'll

have broader definitions in the future but
if you say sine of thirty degrees

hey this ain't gold right over here is thirty
degrees so i can use this right triangle

and we just have to remember soh cah toa

rewrite it so

cah

toa

sine tells us soh tells us what to do with sine. sine is opposite over hypotenuse.

sine of thirty degrees is the opposite side

that is the opposite side which is two

over the hypotenuse. the hypotenuse here is four.

it is two fourths which is the same thing as
one-half

sine of thirty degrees you'll see is always going
to be equal

to one-half

now what is

the cosine

what is the cosine of

thirty degrees

once again go back to soh cah toa.

the cah tells us what to do with cosine. cosine is adjacent over hypotenuse

so for looking at the thirty degree angle
it's the adjacent this right over here is

adjacent it's right next to it

it's not the hypotenuse

it's the adjacent over the hypotenuse so
it's two

square roots of three

adjacent

over

over the hypotenuse over four

or if we simplify that we divide the numerator and the denominator by two it's the square root of three

over two

finally let's do

the tangent

tangent of thirty degrees

we go back to soh cah toa

soh cah toa

toa tells us what to do with tangent
it's opposite over adjacent

you go to the thirty degree angle because that's what we care about, tangent of thirty

tangent of thirty opposite is two

opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent

to it

adjacent means next to

so two square roots of three

so this is equal to

the twos cancel out one over the square root
of three

or we could multiply the numerator and the denominator
by the square root of three

so we have

square root of three

over square root of three

and so this is going to be equal to the numerator
square root of three and then the denominator

right over here is just going to be three so
thats we've rationalized a square root of three

over three

fair enough

now lets use the same triangle to figure out the
trig ratios for the sixty degrees

since we've already drawn it

so what is

what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side

is opposite

what opens out into the two square roots of three
so the opposite side is two square roots of three

and from the sixty degree angle the adj-oh sorry its the

opposite over hypotenuse, don't want to confuse you.

so it is opposite over hypotenuse

so it's two square roots of three over four. four is the hypotenuse.

so it is equal to, this simplifies to square root of three over two.

what is the cosine of sixty degrees. cosine of sixty degrees.

so remember soh cah toa. cosine is adjacent over hypotenuse.

adjacent is the two sides right next to the sixty degree angle so it's two

over the hypotenuse which is four

so this is equal to

one-half

and then finally

what is the tangent, what is the tangent

of sixty degrees

well tangent soh cah toa tangent is opposite
over adjacent

opposite the sixty degrees

is two square roots of three

two square roots of three

and adjacent to that

adjacent to that

is two

adjacent to sixty degrees is two

so its opposite over adjacent

two square roots of three over two which is just equal to

the square root of three

And I just wanted to - look how these are related

the sine of thirty degrees is the same as the cosine of sixty degrees. The cosine of thirty degrees is the same thing as the sine of sixty degrees

and then these guys are the inverse of each other and i think if you think a little bit about this triangle

it will start to make sense why. we'll keep extending
this and give you a lot more practice in the next

few videos