Welcome.
Well what I want to do now is actually I'm going to use this
graphing application to explore the trigonometric functions or
explore the graphs of them.
And just to start off or just to let you, this application
I'm using is from gcalc.net.
it's G C A L C .net.
It's not mine, but I want to give them credit because
that's what I'm using and I hope they don't mind.
So let's start off just graphing some functions.
Let's start off with the sine function.
So let's say sine of x.
I hope you all can see it.
I'm typing it in up here.
So sine x, let me graph that.
Look at that.
Like how nice that looks.
So let's interpret this.
So it's oscillating between-- well, let's
just go point by point.
I guess that's the easiest way to do it.
So when x is equal to 0, what is the value of this function?
Well, if we look here, the value of the function is--
let me actually trace it.
When x equals 0 and it has it written down at the
bottom of this grey area.
So when x is 0, y is also 0.
And if you're remember when we looked for the definitions
in the unit circle that's what we got.
That the sine of 0 radians is 0.
And now as we move on, or move along the curve, I
have the trace function on.
This is when x is equal-- if we look in the grey area at the
bottom left it says 1.57.
But what is that?
If you're familiar with the-- 1.57 is more
commonly known as what?
It's 1/2 of what famous number?
Right.
It's half of pi.
We're at pi over 2.
And if you want to convert pi over 2 to degrees
that's 90 degrees.
So when we're at the angle of pi over 2 radians the sine
function is equal to 1.
And if you go back to some previous modules you'll
remember that that's exactly what the sine function was
equal to when we looked at the unit circle.
Because we were essentially at the point 1 comma 0.
I hope it's not confusing that I keep referring to the unit
circle that you can't see.
But we'll keep going.
But one thing I want to introduce here is the concept
of the period or the frequency of the sine function.
It's pretty obvious to you at this point that the function
keeps repeating itself.
It goes from 0, moves up to 1, goes back to 0, goes
down to negative 1.
Then goes back to 0 and then repeats again.
So the period of this periodic function because that's what we
call a function that keeps repeating, the period of this
periodic function is this distance from here to here.
And what's that?
Well, that's 2 pi radians.
And does that make sense?
Well sure, because 2 pi radians is one complete revolution
around the unit circle.
And then it repeats again.
And then it goes the other way.
You go 2 pi radians backwards and things
start repeating again.
Pretty interesting, right?
Oh, and another thing.
What two numbers does it oscillate between?
It oscillates between positive 1 and negative 1.
And that makes sense because in the unit circle you can never
get to a point on the perimeter of the unit circle that's
larger than positive 1 or less than negative 1.
And that's why the sine of x keeps oscillating between
these two points.
Let's do the cosine of x.
Actually, I'm going to leave the sine of x there.
Interesting.
It looks almost the same, but it looks shifted.
It actually looks shifted to the left about
pi over 2 radians.
And that's actually the case.
So let's first think about why.
We figured out before that sine-- actually, it looks
like this program is still tracing the sine function.
That sine of 0 was 0.
But if you look at the green function, the cosine of
0 radians is actually 1.
Let me see if I can-- no.
I don't know how to trace the cosine function, so
I'll just do it here.
The cosine of 0 is 1.
And why does that make sense?
Well, the cosine is the x-coordinate on
the unit circle.
When you have 0 radians or 0 degrees, you're at the point 1
comma 0 on the unit circle.
So 1 is the cosine or is the x-coordinate and
0 is the sine value.
And if any of this is confusing, review the video
where I use the unit circle to solve the various values of the
trig functions and then this should make sense.
And notice that this has a period similar to
the sine function.
It starts at 1, goes down to negative 1, and then
comes back to positive 1.
And it takes 2 pi radians to complete that cycle.
And just like the sine function it's oscillating between 1 and
negative 1 because on the unit circle you can't get to a point
on the perimeter that's higher than that.
And now to really hit the point home let's do
the tangent function.
I think this one might surprise you.
Well, look at that.
So the blue line is the tangent function.
And why does it do this crazy thing?
Well, if you remember, the tangent function is equal to
the y over the x on the perimeter of the unit circle.
Or since the y is the sine and cosine is the x, it also equals
the sine over the cosine.
So here, tangent is 0 whatever sine is 0
because that makes sense.
Because tangent is equal to sine over cosine.
So it makes sense that when sine is 0, tangent is 0.
But then, as the sine function becomes greater and the cosine
function becomes less, the numerator in the tangent
function becomes greater because the numerator is sine.
So we get larger and larger values, all the way to the
point where the denominator of the tangent function, which is
the cosine function-- I think this is probably the most
confusing module I've ever said because I can't really
write these things down.
The denominator goes to 0.
The cosine right here.
And then tan spikes and it actually approaches infinity.
And if you look back at the unit circle, that actually
might make a little bit of sense.
But like the other functions, actually the tangent
function has a period of pi instead of pi over 2.
Instead of 2 pi.
And I'll leave that as an exercise for you
to think about.
But with that drawn out, I'm now going to
explore something else.
Let me reset this.
Yes, I really want to reset.
I drew the sine function before.
Let me draw the sine of let's say, 2x.
Whoops, that's not right.
sine of 2-- maybe I need to put some parentheses in.
Oh, there we go.
Actually, let me reset it.
Yes, I want to reset.
So first I'll draw the sine of x and then I'll
draw the sine of 2x.
So what's the first thing you notice about the difference
between these two?
The brown one is sine of x and the green one is sine of 2x.
They both oscillate between the same two numbers and just so
you know, the height of the oscillation is called
the amplitude of this periodic function.
So in both cases, the amplitude is 1 because they oscillate
from 1 to negative 1.
So the amplitude is 1, but their period is different.
Sine of x takes 2 pi radians to complete one circle-- one cycle
while sine of 2x only takes pi radians to complete one cycle.
So it actually completes it twice as fast.
And I want you to sit and think about why sine of 2x has 1/2
the period of sine of x.
And you can probably guess what happens if
I type in sine of 3x.
Actually, let's do sine of 4x.
It should have 1/2 the period of sine of 2x then.
And it does, even though this is probably a
very confusing graph.
So let's explore.
So I think you understand what the coefficient
on the x term does.
When you have a larger coefficient it kind of
speeds up the cycles.
And let's explore a little bit more.
Let's start off with sine of x again.
And now, instead of making the coefficient larger, let's
make the coefficient less.
Let's make it sine of 0.5x.
Look at that.
Now, all of a sudden, it takes 4 pi radians to
complete one cycle.
And I want you to think about why that is.
Because we're now slowing down how fast it cycles
through the angles.
Now I want to start playing with the amplitude.
So we had sine of x, what do you think will happen if I put
in this 2 times sine of x?
So here, the period is the same.
It's still 2 pi, but notice that it oscillates between 2
and negative 2 instead of between 1 and negative 1.
So whatever the coefficient, or whatever the number is in front
of the sine or the cosine function, that actually
affects its amplitude.
And similarly, we can look at 0.5 sine-- let's
say 0.5 sine of 2x.
Interesting.
So now it only goes up to 0.5 and down to minus 0.5.
So it's amplitude is 1/2 or 0.5.
And it also oscillates twice as fast as the sine function
because it was 0.5 sine of 2z.
I think that's all the time I have now.
I have a feeling this might have confused you more than
helped, but I'll still put the video up just in case
it's helpful for someone.
But in the future I might actually record another video
where I can actually write things down so it doesn't
confuse you as much.
So if it confused you I apologize, but I
hope it was helpful.
See you later.