Hello.
In the last presentation we kind of re-defined the sine,
the cosine, and the tangent functions in a broader way
where we said if we have a unit circle and our theta is, or our
angle, is -- let me use the right tool -- let's say, and
our angle is the angle between, say, the x-axis and a radius in
the unit circle, and this is our radius.
the coordinate of the point where this radius intersects
the unit circle is x comma y.
Our new definition of the trig functions was that sine of
theta is equal to the y-coordinate, right, this is
y-coordinate where it intersects the unit circle.
And remember, this is the unit circle.
It's not just any circle, which means it has a radius of 1.
Cosine of theta is equal to the x-coordinate of this point.
This is the x-coordinate.
And tangent of theta equaled opposite over
adjacent or y over x.
That's interesting because that's also equal to sine of
theta over cosine of theta.
I'll just do that.
I wasn't even planning on covering that, but just it
leaves you something to think about.
So with that said, let's take a look or let's try to see how
this defines these functions.
And I guess a good place to start is just with the sine
function and we can try to graph it.
So let's write, let's do a little table like we always do
when we define a function.
Let's put in values of theta, and let's figure
out what sine of theta is.
So when theta is equal to 0 radians, what is sine of theta?
So when theta's 0, right, then the radius between it -- this
is the radius and this is the point where the radius
intersects the unit circle.
And this point has a coordinate 1 comma 0, right?
And so if where it intersects the unit circle is at 1 comma
0, then sine of theta is just the y-coordinate.
So sine of theta is 0.
If we said what is sine of theta when theta is
equal to pi over 2.
So now our radius is this radius and we intersect the
unit circle right here at the point 0 comma 1.
And what's the y-coordinate at 0 comma 1?
Well it's 1.
What happens when we have theta is equal to pi radians?
So at pi radians we intersect the unit circle right here.
We're at pi radian.
This is the angle, pi.
We intersect with unit circle at negative 1 comma 0.
Because once again, this is the unit circle.
So at negative 1 comma 0, what's the y-coordinate?
Well, it's 0.
So sine of pi is equal to 0.
Let's just keep going around the circle.
When we have the angle, when theta is equal to 3 pi over 4
-- no, sorry, 3 pi over 2.
Because this is pi and this is another half pi.
So this is 3 pi over 2, sorry.
So when theta is equal to 3 pi over 2, what is sine of theta?
Well, now we intersect the unit circle down here at the
point 0 comma negative 1.
So now sine of theta is equal to negative 1.
Then if we go all the way around the circle to 2
pi radians, we're back at this point again.
So sine of theta, so we're at 2 pi, sine of theta
is now 0 once again.
So let's graph these points out and then we'll try to figure
out what the points in between look like, and I'll show
you the graph of a sine function is.
So let's draw the x-axis.
This is my x-axis.
And let's draw the y-axis.
Not as clean as I wanted to draw it.
This is y.
And that's x.
But in this case instead of saying that's the x-axis, let's
call that the theta axis, because we defined theta as the
input or our domains in terms of theta.
So this is the theta axis.
Now we're going to graph sine of theta.
So when we said when theta equaled 0, sine
of theta is equal to 0.
So that's this point right here, 0 comma 0.
When theta is equal to pi over 2, sine of theta is equal to 1.
So this is the point pi over 2 comma 1, right?
That's just this 1.
When theta is equal to pi, sine of theta is 0 again.
So this is the point pi comma 0.
And when theta equaled 3 pi over 2, what was sine of theta?
I equaled negative 1.
Interesting.
Then when we got to 2 pi -- when we got to theta equal
to 2 pi, sine of theta, again, equaled 0.
So we know that these points are on the graph
of sine of theta.
And if you actually tried the points in between, and as
an exercise it might be interesting for you to do so.
You could actually figure out a lot of the points using
30-60-90 triangles or using the Pythagorean Theorem.
But you actually get a curve that looks something -- let me
use a nicer color than this kind of drab grey -- you
get a graph that looks something like this.
And you've probably seen that before.
The term for this function is actually a sine wave.
It looks like something that's oscillating or
that's moving up and down.
And actually if you were to put in thetas that were less than
0, the sine wave will keep going into the
negative theta axis.
It keeps going forever in both directions.
It keeps oscillating between 1 and negative 1 and
the points in between.
So that's the graph of the sign function.
In the next module I'll actually do the graph of the
cosine function, or actually I might just show you the graph
of the cosine function.
Then I'll show you how they relate and how these can
describe any kind of, or many types of oscillatory things in
the world and how it relates to frequency and amplitude.
So I'll see you in the next module.
And just for fun you might want to sit down with a piece of
paper and try to graph the cosine function or the
tangent function as well.
Have fun.