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 will only work in right triangles, so if you try to find
the trig
functions of angles that aren't part of right triangles, we're goint to see a 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 - the sum of the squares
of the other two sides. Eight squared is equal to seven squared plus four squared.
So this is equal to forty-nine
plus sixteen,
fourty-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 right: 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 see 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 is 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 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 about
some type of indian princess named "soh cah toa" or whatever, but it's a very useful
pneumonic, so we can apply "soh cah toa" to 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" 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, it's 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 the
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 denominator square two sixty five times square to 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 will learn in the future that there's a ton of them
but they're all derived from these
so let's think about 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 is
right here that is the opposite side
and then in the
hypotenuse, it's opposite over hypotenuse. the hypotenuse is the
square root sixty-five
and once again if we want to rationalize this we could multiply times the square root of sixty- five
over the square root of sixty-five
and the numerator will get seven square roots 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
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
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