We've got two right triangles here.
And let's say we also know that they both have
an angle whose measure is equal to theta.
So angle A is congruent to angle D. What
do we now know about these two triangles?
Well for any triangle, if you know two of the angles,
you're going to know the third angle,
because the sum of the angles of a triangle
add up to 180 degrees.
So if you have two angles in common,
that means you're going to have three angles in common.
And if you have three angles in common,
you are dealing with similar triangles.
Let me make that a little bit clearer.
So if this angle is theta, this is 90.
They all have to add up to 180 degrees.
That means that this angle plus this angle
up here have to add up to 90.
We've already used up 90 right over here,
so angle A and angle B need to be complements.
So this angle right over here needs to be 90 minus theta.
Well we could use the same logic over here.
We already use of 90 degrees over here.
So we have a remaining 90 degrees
between theta and that angle.
So this angle is going to be 90 degrees minus theta.
You have three corresponding angles being congruent.
You are dealing with similar triangles.
Now why is that interesting?
Well we know from geometry that the ratio
of corresponding sides of similar triangles
are always going to be the same.
So let's explore the corresponding sides here.
Well, the side that jumps out-- when
you're dealing with the right triangles-- the most
is always the hypotenuse.
So this right over here is the hypotenuse.
This hypotenuse is going to correspond
to this hypotenuse right over here.
And then we could write that down.
This is the hypotenuse of this triangle.
This is the hypotenuse of that triangle.
Now this side right over here, side BC,
what side does that correspond to?
Well if you look at this triangle,
you can view it as the side that is opposite this angle theta.
So it's opposite.
If you go across the triangle, you get there.
So let's go opposite angle D.
If you go opposite angle A, you get to BC.
Opposite angle D, you get to EF.
So it corresponds to this side right over here.
And then finally, side AC is the one remaining one.
We could view it as, well, there's
two sides that make up this angle A. One of them
is the hypotenuse.
We could call this, maybe, the adjacent side to it.
And so D corresponds to A, and so this
would be the side that corresponds.
Now the whole reason I did that is to leverage
that, corresponding sides, the ratio
between corresponding sides of similar triangles,
is always going to be the same.
So for example, the ratio between BC and the hypotenuse,
BA-- so let me write that down-- BC/BA
is going to be equal to EF/ED, the length of segment EF
over the length of segment ED.
Or we could also write that the length
of segment AC over the hypotenuse,
over this triangle's hypotenuse, over AB,
is equal to DF/DE-- once again, this green side
over the orange side.
These are similar triangles.
They're corresponding to each other.
So this is equal to DF/DE.
And we could keep going, but I'll just do another one.
Or we could say that the ratio of this side right
over here-- this blue side to the green side
of this triangle-- the length of BC/CA
is going to be the same as the ratio between these two
corresponding sides, the blue over the green, EF/DF.
And we got all of this from the fact
that these are similar triangles.
So this is true for any right triangle
that has an angle theta.
Then those two triangles are going to be similar,
and all of these ratios are going to be the same.
Well, maybe we can give names to these ratios
relative to the angle theta.
So from angle theta's point of view--
I'll write theta right over here,
or we can just remember that-- what
is the ratio of these two sides?
Well from theta's point of view, that blue side
is the opposite side.
It's opposite-- so the opposite side of the right triangle.
And then the orange side we've already labeled the hypotenuse.
So from theta's point of view, this
is the opposite side over the hypotenuse.
And I keep stating from theta's point of view
because that wouldn't be the case for this other angle,
for angle B. From angle B's point of view,
this is the adjacent side over the hypotenuse.
And we'll think about that relationship later on.
But let's just all think of it from theta's point
of view right over here.
So from theta's point of view, what is this?
Well theta's right over here.
Clearly AB and DE are still the hypotenuses-- hypoteni.
I don't know how to say that in plural again.
And what is AC, and what are DF?
Well, these are adjacent to it.
They're one of the two sides that
make up this angle that is not the hypotenuse.
So this we can view as the ratio, in either
of these triangles, between the adjacent side--
so this is relative.
Once again, this is opposite angle B,
but we're only thinking about angle
A right here, or the angle that measures theta,
or angle D right over here-- relative to angle A,
AC is adjacent.
Relative to angle D, DF is adjacent.
So this ratio right over here is the adjacent
over the hypotenuse.
And it's going to be the same for any right triangle that
has an angle theta in it.
And then finally, this over here, this
is going to be the opposite side.
Once again, this was the opposite side over here.
This ratio for either right triangle
is going to be the opposite side over the adjacent side.
And I really want to stress the importance--
and we're going to do many, many more examples of this
to make this very concrete-- but for any right triangle
that has an angle theta, the ratio between its opposite side
and its hypotenuse is going to be the same.
That comes out of similar triangles.
We've just explored that.
The ratio between the adjacent side
to that angle that is theta and the hypotenuse
is going to be the same, for any of these triangles,
as long as it has that angle theta in it.
And the ratio, relative to the angle
theta, between the opposite side and the adjacent side,
between the blue side and the green side,
is always going to be the same.
These are similar triangles.
So given that, mathematicians decided
to give these things names.
Relative to the angle theta, this ratio
is always going to be the same, so the opposite
over hypotenuse, they call this the sine of the angle theta.
Let me do this in a new color-- by definition--
and we're going to extend this definition in the future--
this is sine of theta.
This right over here, by definition,
is the cosine of theta.
And this right over here, by definition,
is the tangent of theta.
And a mnemonic that will help you remember this-- and these
really are just definitions.
People realized, wow, by similar triangles, for any angle theta,
this ratio is always going to be the same.
Because of similar triangles, for any angle theta,
this ratio is always going to be the same.
This ratio is always going to the same.
So let's make these definitions.
And to help us remember it, there's
the mnemonic soh-cah-toa.
So I'll write it like this.
soh is sine is opposite over hypotenuse.
cah-- cosine is adjacent over hypotenuse.
And then finally, tangent is opposite
over adjacent-- soh-cah-toa.
And in future videos, we'll actually
apply these definitions for these trigonometric functions.