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 lengthy 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 the equal to forty-nine
plus sixteen,
fourty-nine plus sixteen,
forty nine plus ten is fifty-nine, plus
six is
sixty-five. It is a 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 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
new monic, 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 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 trick functions
carelessly said the court rick functions will
learn in the future that there's a few tumble
but they're all derived from these
fussing about the sign of state is once again
to go to so cut off
the sole tells what to do it signed side of
the opposite over high positives
sign
is equal to
opposite over hypothesize opposite over high
partners
so for this and all tried as officer
we just go
opposite itworld 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
hypothalamus opposite overabundance of hypothesis
in the square root of sixty-five
square root sixty-five
and once again if we want to rationalize this
week a multiply times square in sixty five
over the square to sixty-five
numerator will get seven square roots of sixty-five
and in the denominator we will get jest
sixty-five again
knowledge do tangent
let us do ten gente
so if i ask you the tangent
but ended up there
once again go back to soak up
toa the toll apart 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 ed to use it
so for this and all
what is opposite we've already figured it
out it's evident opens into the seventh opposite
the seven
so it's seven
over website is adjacent
well this for is adjacent
this for is adjacent to the jason side is
four
so it's seven
over
and were dot
we figured out all of the trade ratios for
baidoa let's do it other one
let's do another one on the colluded concrete
'cause right now we've been saying oh was
tentative actress tens of their lives make
a little bit more concrete
let's say
let's say i mean drawn other right triangle
list another right triangle here
everything we're dealing with
these are going to be right triangles
let's say the high partners
has the length
let's say that the side over here
has length too
and let's say that this length overhears will
be to times the square root of three we can
verify that this works
if you have this aside squared so yet let
me write down two times the square root of
three squared
plus to squared is equal to what
this is
to there's going to be four times three
four times three plus four
and this is going to be equal to twelve plus
for the for the sixteen and sixteen is indeed
four square so this does he call for squared
it does equal foursquare it satisfies the
protagonist hero
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 eight 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 the right triangle
this angle right over here is our thirty degree
angle
and then disable appear the single appears
is
sixty degree angle
and it's a pretty sixteen ninety because
the side opposite the thirty degrees is half
of my partners
and in the side opposite the sixty degrees
is a square to three times the other side
that's not the hype on this
so that's that we're not gonna the student
was peer review of thirty sixty ninety triangles
although i just get it
let's actually fifi and the trigger issues
for the different angles
so if i were to ask you
or if anyone would ask you what is
what is the signed of thirty degrees
and remember thirty degrees is one of the
eagles in this triangle but it would apply
whenever you have a thirty degree angle and
you're dealing with the right rangel will
have broader definitions of the future but
if you see a sign of thirty degrees
hey this ain't gold right over here is thirty
degrees so i can use this right triangle
and we should remember so catawba
rewrite it so
ka
toa
sign tells us more also tells us what to do
it signed side is opposite over hype rotten
is
sign up thirty degrees is the opposite side
that is the opposite side which is to
over the hard part is the hype on this year's
for
it is to ford's which is the same thing as
one-half
five thirty degrees you'll see is always going
to be equal
to one-half
now what is
the duck cosign
what is the co-signed of
thirty degrees
once again go back to seoul cut off
the cod tells us what to do with co-sign co-sign
isn't jay sent over high popped in his
so for looking at the thirty degree angle
it's the end jason this right over here is
adjacent it's right next to it
it's not my pot news
it's the adjacent over the high positive so
it's too
square roots of three
and jay sent
over
over that partners over four
or if we simplify that we divide the new renowned
abide to this the square root of three
over too
finally let's do that ended
that tangent
thirty degrees
we go back to soak up
draw
so cut off poet tells us what to do it entered
it's opposite over adjacent
you go to the thirty degree angle to the so
we care about ten to twelve thirty
tended of thirty opposite is too
opposite is to and the adjacent is to squirts
of three it's right next to it it's adjacent
to it
and jason means next to
so to square roots of three
so this is equal to
the tunes cancel out one over the square root
of three
oregon multiplied the numerator denominator
by the square 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 the story and then the denominator
right overhear is just going to be three so
thats we've rationalize a square two three
over three
fair enough
welch is the same triangle figure out the
tree gracias for the sixty degrees
century we've already brought it
so what is
what is in the sign of the sixty degrees and
interval for getting the hang of it now
side is opposite over js insult from the sock
it all out for the sixty degree angle website
is opposite
what opens out into the two squares of three
so the opposite side is two squares history
from the sixty degree angle dj course ties
observer had taught me a story confuse you
so it's opposite over high pot and his sister
squirts of three over four force a high partners
so it is in equal to the simplifies to square
root of three
over too
what is the the co-sign a sixty degrees
co-sign of sixty degrees
so remember so cut off co-sign is adjacent
over my pot news
it jason's is the two sides right next to
sixty degree angle so it's too
over the hype on news which is four
so this is equal to
one-half
and then finally
what is a bit engine
what is the tangent
of sixty degrees
well tended so cut toa attended is opposite
over adjacent
opposite the sixty degrees
is to squirts of three
to squirts of three
and adjacent to that
in j sent to that
adjacent to sixty degrees is too so it's
opposite over it isn't too squirts of three
over too
which is just equal to the square read of
and i just wanted you look at these are related
to sign a third agrees is the same things
of course i love sixty degrees
because i'm a third agrees with the same thing
as a sign of sixty degrees and that these
guys are the inverse of each other i think
if you think of a little bit about the strike
both
will start to make sense why will keep extending
this may be a lot more practice in the next
few videos