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Let's just do a ton of more examples, just so we
make sure that we're getting
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this trig function thing down well.
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So let's construct ourselves some right triangles.
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Let's construct ourselves some right triangles, and I want to be very clear.
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The way I've defined it so far, this will only work in right triangles.
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So if you're trying to find the trig functions of angles that aren't part of right triangles,
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we're going to see that we're going to have to construct right triangles,
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but let's just focus on the right triangles for now.
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So let's say that I have a triangle, where
let's say this length down here is seven,
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and let's say the length of this side up here, let's say that that is four.
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Let's figure out what the hypotenuse over here is going to be.
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So we know -let's call the hypotenuse, "h"-
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we know that h squared is going to be equal
to seven squared plus four squared,
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we know that from the Pythagorean theorem,
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that the hypotenuse squared is equal to
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the square of each of the sum of the squares of the other two sides.
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h squared is equal to seven
squared plus four squared.
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So this is equal to forty-nine plus sixteen,
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forty-nine plus sixteen,
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forty nine plus ten is fifty-nine, plus
six is sixty-five.
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It is sixty five. So this h squared,
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let me write: h squared -that's different shade of yellow-
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so we have h squared is equal to sixty-five.
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Did I do that right? Forty nine plus ten is fifty nine, plus another six is sixty-five,
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or we could say that h is equal to, if we take the square root of both sides,
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square root
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square root of sixty five. And we really can't simplify
this at all.
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This is thirteen.
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This is the same thing as thirteen times five,
both of those are not perfect squares and
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they're both prime so you can't simplify this any more.
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So this is equal to the square root of sixty five.
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Now let's find the trig, let's find the trig functions for this angle up here.
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Let's call that angle up there theta.
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So whenever you do it
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you always want to write down - at least for
me it works out to write down -
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"soh cah toa".
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soh...
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...soh cah toa. I have these vague memories
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of my trigonometry teacher.
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Maybe I've read it in some book. I don't know - you know, some... about
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some type of indian princess named "soh cah toa" or whatever,
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but it's a very useful mnemonic,
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so we can apply "soh cah toa".
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Let's find, let's say we want to find the cosine.
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We want to find the cosine of our angle.
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We wanna find the cosine of our angle, you
say: "soh cah toa!"
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So the "cah". "Cah" tells us what to do with cosine,
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the "cah" part tells us
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that cosine is adjacent over hypotenuse.
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Cosine is equal to adjacent over hypotenuse.
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So let's look over here to theta; what side is adjacent?
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Well we know that the hypotenuse,
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we know that that hypotenuse is this side over here.
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So it can't be that side. The only other side that's kind of adjacent to it that
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isn't the hypotenuse, is this four.
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So the adjacent side over here, that side is,
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it's literally right next to the angle, it's one of
the sides that kind of forms the angle
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it's four over the hypotenuse.
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The hypotenuse we already know is square root
of sixty-five.
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so it's four over the square root of sixty-five.
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And sometimes people will want you to rationalize
the denominator which means
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they don't like to have an irrational number in the denominator,
like the square root of sixty five,
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and if they - if you wanna rewrite this without
a irrational number in the denominator,
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you can
multiply the numerator and the denominator
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by the square root of sixty-five.
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This clearly will not change the number, because we're multiplying it by something over itself,
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so we're multiplying the number by one.
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That won't change the number, but at least it gets rid of the irrational number in the denominator.
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So the numerator
becomes
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four times the square root of sixty-five,
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and the denominator, square root of sixty five times
square root of sixty-five, is just going to be sixty-five.
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We didn't get rid of the irrational number, it's still
there, but it's now in the numerator.
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Now let's do the other trig functions
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or at least the other core trig functions. We'll
learn in the future that there's actually a ton of them
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but they're all derived from these.
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so let's think about what the sign of theta is. Once again
go to "soh cah toa".
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The "soh" tells what to do with sine. Sine is opposite over hypotenuse.
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Sine is equal to opposite over hypotenuse.
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Sine is opposite over hypotenuse.
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So for this angle what side is opposite?
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We just go opposite it, what it opens into, it's opposite
the seven
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so the opposite side is the seven.
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This is, right here - that is the opposite side
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and then the hypotenuse, it's opposite over hypotenuse.
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The hypotenuse is the square root of sixty-five.
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Square root of sixty-five.
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and once again if we wanted to rationalize this,
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we could multiply times the square root of sixty-five over the square root of sixty-five
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and the the numerator, we will get seven square root of sixty-five
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and in the denominator we will get just sixty-five again.
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Now let's do tangent!
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Let us do tangent.
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So if i ask you the tangent
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of - the tangent of theta
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once again go back to "soh cah toa".
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The toa part tells us what to do with tangent
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it tells us...
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it tells us that tangent
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is equal to opposite over adjacent
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is equal to opposite over
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opposite over adjacent
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So for this angle, what is opposite? We've already figured it
out.
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it's seven. It opens into the seven.
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It is opposite the seven.
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So it's seven over what side is adjacent.
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well this four is adjacent.
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This four is adjacent. So the adjacent side is
four.
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so it's seven over four,
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and we're done.
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We figured out all of the trig ratios for
theta. let's do another one.
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Let's do another one. i'll make it a little bit concrete
'cause right now we've been saying,
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"oh, what's tangent of x, tangent of theta." let's make it a little bit more concrete.
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Let's say...
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let's say, let me draw another right triangle,
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that's another right triangle here.
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Everything we're dealing with, these are going to be right triangles.
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let's say the hypotenuse has length four,
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let's say that this side over here has length two,
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and let's say that this length over here is going to be two times the square root of three.
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We can verify that this works.
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If you have this side squared, so you have - let
me write it down -
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two times the square root of three squared
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plus two squared, is equal to what?
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this is two. There's going to be four times three.
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four times three plus four,
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and this is going to be equal to twelve plus
four is equal to sixteen
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and sixteen is indeed four squared. So this does equal four squared,
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it does equal four squared. It satisfies the pythagorean theorem
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and if you remember some of your work from thirty
sixty ninety triangles
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that you might have learned in geometry,
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you might recognize that this is a thirty sixty ninety triangle.
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This
right here is our right angle,
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- i should have drawn it from the get go to show that this
is a right triangle -
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this angle right over here is our thirty degree
angle
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and then this angle up here, this angle up here
is
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a sixty degree angle,
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and it's a thirty sixteen ninety because
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the side opposite the thirty degrees is half the hypotenuse
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and then the side opposite the sixty degrees
is a squared of three times the other side
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that's not the hypotenuse.
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So that said, we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles although i just did it.
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Let's actually find the trig ratios for the different angles.
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So if i were to ask you or if anyone were to ask you, what is...
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what is the sine of thirty degrees?
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and remember thirty degrees is one of the
angles in this triangle but it would apply
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whenever you have a thirty degree angle and
you're dealing with the right triangle.
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We'll have broader definitions in the future but
if you say sine of thirty degrees,
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hey, this angle right over here is thirty
degrees so i can use this right triangle,
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and we just have to remember "soh cah toa"
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We rewrite it. soh, cah, toa.
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"sine tells us" (correction). soh tells us what to do with sine. sine is opposite over hypotenuse.
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sine of thirty degrees is the opposite side,
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that is the opposite side which is two over the hypotenuse.
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The hypotenuse here is four.
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it is two fourths which is the same thing as
one-half.
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sine of thirty degrees you'll see is always going
to be equal to one-half.
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now what is the cosine?
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What is the cosine of thirty degrees?
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Once again go back to "soh cah toa".
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The cah tells us what to do with cosine.
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Cosine is adjacent over hypotenuse.
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So for looking at the thirty degree angle
it's the adjacent.
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This, right over here is adjacent. it's right next to it.
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it's not the hypotenuse. it's the adjacent over the hypotenuse.
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so it's two square roots of three
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adjacent over...over the hypotenuse, over four.
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or if we simplify that, we divide the numerator and the denominator by two
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it's the square root of three over two.
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Finally, let's do the tangent.
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The tangent of thirty degrees,
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we go back to "soh cah toa".
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soh cah toa
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toa tells us what to do with tangent. It's opposite over adjacent
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you go to the thirty degree angle because that's what we care about, tangent of thirty.
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tangent of thirty. Opposite is two,
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opposite is two and the adjacent is two square roots of three.
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It's right next to it it's adjacent to it.
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adjacent means next to.
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so two square roots of three
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so this is equal to... the twos cancel out
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one over the square root
of three
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or we could multiply the numerator and the denominator
by the square root of three.
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So we have square root of three over square root of three
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and so this is going to be equal to the numerator
square root of three and then
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the denominator right over here is just going to be three.
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So that we've rationalized a square root of three over three.
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Fair enough.
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Now lets use the same triangle to figure out the
trig ratios for the sixty degrees,
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since we've already drawn it.
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so what is... what is the sine of the sixty degrees?
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and i think you're hopefully getting the hang of it now.
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Sine is opposite over adjacent. soh from the "soh cah toa".
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for the sixty degree angle what side is opposite?
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what opens out into the two square roots of three,
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so the opposite side is two square roots of three,
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and from the sixty degree angle the adj-oh sorry
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its the opposite over hypotenuse, don't want to confuse you.
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so it is opposite over hypotenuse
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so it's two square roots of three over four. four is the hypotenuse.
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so it is equal to, this simplifies to square root of three over two.
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What is the cosine of sixty degrees? cosine of sixty degrees.
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so remember "soh cah toa". cosine is adjacent over hypotenuse.
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adjacent is the two sides, right next to the sixty degree angle.
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So it's two over the hypotenuse which is four.
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So this is equal to one-half
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and then finally, what is the tangent?
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what is the tangent of sixty degrees?
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Well tangent, "soh cah toa". Tangent is opposite
over adjacent
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opposite the sixty degrees
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is two square roots of three
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two square roots of three
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and adjacent to that
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adjacent to that is two.
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Adjacent to sixty degrees is two.
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So its opposite over adjacent, two square roots of three over two
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which is just equal to the square root of three.
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And I just wanted to -look how these are related-
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the sine of thirty degrees is the same as the cosine of sixty degrees.
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The cosine of thirty degrees is the same thing as the sine of sixty degrees
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and then these guys are the inverse of each other and i think if you think a little bit about this triangle
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it will start to make sense why.
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we'll keep extending
this and give you a lot more practice in the next few videos.