WEBVTT mathematics/trigonometry/murray
00:00:00.000 --> 00:00:07.000
Hi, this is educator.com and we are here to talk about sine and cosine functions.
00:00:07.000 --> 00:00:10.000
I'll start out by giving you the definitions and kind of the master formulas.
00:00:10.000 --> 00:00:15.000
Then we'll go through and work on a bunch of examples.
00:00:15.000 --> 00:00:29.000
The definition of sine and cosine of an angle is, you start out with the axis and the unit circle it's important to know that.
00:00:29.000 --> 00:00:33.000
This is a unit circle meaning the radius is 1.
00:00:33.000 --> 00:00:42.000
What you do is you draw that angle in standard position, meaning it has one side on the x-axis.
00:00:42.000 --> 00:00:44.000
There is data right there.
00:00:44.000 --> 00:00:51.000
Then you look at the coordinates of the point, the x and y coordinates on the unit circle.
00:00:51.000 --> 00:01:00.000
The x-coordinate is, I'll go over that in red, that's the x coordinate right there.
00:01:00.000 --> 00:01:03.000
The y-coordinate, that's the one in blue.
00:01:03.000 --> 00:01:11.000
Those coordinates are, by definition, the...
00:01:11.000 --> 00:01:23.000
I want to do the cosine, in red, cos(θ) and then, in blue, sin(θ).
00:01:23.000 --> 00:01:30.000
That's the definition of what an angle is in terms of coordinates on the unit circle.
00:01:30.000 --> 00:01:37.000
If you know what the angle is, you try to figure out what its x-coordinate and its y-coordinate are.
00:01:37.000 --> 00:01:41.000
Let's call these cos(θ) and the sin(θ).
00:01:41.000 --> 00:01:53.000
Now, we'll see some more examples of that later so that you'll know how to actually compute the cosine and sine, but the definition just refers to those coordinates.
00:01:53.000 --> 00:02:00.000
The most common use of sine and cosine probably is in terms of right triangles.
00:02:00.000 --> 00:02:03.000
Let me draw a right triangle.
00:02:03.000 --> 00:02:10.000
Right triangle just means a triangle where one of the angles is a right angle, a 90-degree angle, or in terms of radians, π over 2.
00:02:10.000 --> 00:02:19.000
What you do is, you let θ be one of the angles that is not the 90-degree angle, so one of the other angles.
00:02:19.000 --> 00:02:25.000
Then you label each one of the sides in terms of its relationship to θ.
00:02:25.000 --> 00:02:31.000
The one next to θ is called the adjacent side.
00:02:31.000 --> 00:02:41.000
The one opposite θ is called the opposite side.
00:02:41.000 --> 00:02:48.000
The long side is called the hypotenuse.
00:02:48.000 --> 00:03:01.000
Then we have the master formula for right triangles, which is, the sine of θ is equal to the length of the opposite side divided by the hypotenuse.
00:03:01.000 --> 00:03:05.000
The cosine of θ is equal to the adjacent side divided by the hypotenuse.
00:03:05.000 --> 00:03:13.000
The tangent of θ, which is something we haven't officially defined yet, so we'll learn about tangent in a later lesson.
00:03:13.000 --> 00:03:20.000
I just want to give you the right triangle formula now, because we're going to try to remember them all together.
00:03:20.000 --> 00:03:25.000
The tangent of θ when we get to it will be the opposite side divided by the adjacent side.
00:03:25.000 --> 00:03:29.000
We don't want to worry too much about tangent now because we haven't learned about it in detail yet.
00:03:29.000 --> 00:03:31.000
I'll get to those later.
00:03:31.000 --> 00:03:37.000
If you put all these formulas together, it's kind of hard to remember their relationships.
00:03:37.000 --> 00:03:41.000
So people have come up with this acronym.
00:03:41.000 --> 00:03:45.000
Sine is equal to opposite over hypotenuse.
00:03:45.000 --> 00:03:47.000
Cosine is equal to adjacent over hypotenuse.
00:03:47.000 --> 00:03:53.000
Tangent is equal to opposite over adjacent.
00:03:53.000 --> 00:03:55.000
If you kind of read that quickly, people call it SOH CAH TOA.
00:03:55.000 --> 00:04:02.000
If you talk to any trigonometry teacher in the world, or any trigonometry student in the world, they should have heard the word SOH CAH TOA,
00:04:02.000 --> 00:04:07.000
because that's kind of the master formula that helps you remember all these relationships.
00:04:07.000 --> 00:04:13.000
They're kind of hard to remember on their own, but if you remember SOH CAH TOA, you won't go wrong.
00:04:13.000 --> 00:04:22.000
If you have trouble remembering that, there is a little mnemonic device that people also use.
00:04:22.000 --> 00:04:28.000
Some old horse caught another horse taking oats away.
00:04:28.000 --> 00:04:36.000
If you remember that sentence, if that's easier for you to remember than SOH CAH TOA, then you can remember all these formulas.
00:04:36.000 --> 00:04:49.000
There is another definition that we need to learn which is, that a function is odd if f(-x) is equal to -f(x).
00:04:49.000 --> 00:04:51.000
Let's figure out what that means.
00:04:51.000 --> 00:04:54.000
We're going to talk about odd and even function.
00:04:54.000 --> 00:04:59.000
Let me give you an example here.
00:04:59.000 --> 00:05:05.000
F(x) is equal to x³.
00:05:05.000 --> 00:05:07.000
Well, let's try f(-x) here.
00:05:07.000 --> 00:05:10.000
I'm going to check this definition of odd function here.
00:05:10.000 --> 00:05:26.000
So, f(-x), that means you put -x into the function, so that's (-x)³ which is (1)3 times x³,
00:05:26.000 --> 00:05:34.000
(-1)³ is -1, so that's just -x³, and that's negative of the original f(x).
00:05:34.000 --> 00:05:42.000
For x³, f(-x) is equal to -f(x), which means it's an odd function.
00:05:42.000 --> 00:05:48.000
There is a way to check this graphically.
00:05:48.000 --> 00:05:55.000
If you graph f(x)=x³, it looks something like this.
00:05:55.000 --> 00:06:19.000
That means that if you look at a particular value of x, and you look at f(x) there, and then you look at -x, f(-x)=-f(x).
00:06:19.000 --> 00:06:23.000
That's what it looks like graphically.
00:06:23.000 --> 00:06:33.000
What it means is that the graph has what I call rotational symmetry around the origin.
00:06:33.000 --> 00:06:40.000
If you put a big dot on the origin and if you spun this graph around 180 degrees, it would look the same.
00:06:40.000 --> 00:06:48.000
That's because f(x) and f(-x) being opposites of each other.
00:06:48.000 --> 00:06:57.000
If you spin the graph around 180 degrees and it looks the same, if it looks symmetric with itself, that's called an odd function.
00:06:57.000 --> 00:07:08.000
The way you remember that odd functions have that property is, just remember x³, x to the third, because 3 is an odd number and x³ is an odd function.
00:07:08.000 --> 00:07:15.000
Something, kind of, has that property that x³ has, then it's an odd function.
00:07:15.000 --> 00:07:23.000
They're companion definition to that is that f is even if f(-x) equals f(x).
00:07:23.000 --> 00:07:27.000
The difference there was that negative sign on the odd definition.
00:07:27.000 --> 00:07:30.000
No negative sign here on the even definition.
00:07:30.000 --> 00:07:36.000
Let me give you an example of an even one.
00:07:36.000 --> 00:07:42.000
Let's define f(x) to be x².
00:07:42.000 --> 00:07:58.000
Well, f(-x) is equal to, you plugin -x into the function, so (-x)² well that's just the same as x², which is the same as the original f(x).
00:07:58.000 --> 00:08:06.000
So, f(-x) is equal to f(x), that checks the definition, so it's even.
00:08:06.000 --> 00:08:14.000
Of course you'll notice that x², the 2 there is an even number, that's no coincidence.
00:08:14.000 --> 00:08:20.000
That's why we call even functions even is because they sort of behave like x².
00:08:20.000 --> 00:08:25.000
If you graph those, let me graph x² for you.
00:08:25.000 --> 00:08:31.000
That's a familiar parabola that you learned how to graph in the algebra section.
00:08:31.000 --> 00:08:37.000
If you take a value of x, and look at f(x),
00:08:37.000 --> 00:08:52.000
then you take f(-x), f(-x) is not -f(x), it's f(x) itself.
00:08:52.000 --> 00:08:57.000
It's the same value as f(x) itself.
00:08:57.000 --> 00:09:02.000
You get this f(-x) is equal to f(x).
00:09:02.000 --> 00:09:15.000
What that means is that you have a, kind of, symmetry across the y-axis with even functions.
00:09:15.000 --> 00:09:27.000
If a function is symmetric across the y-axis, if it looks like a mirror image of itself across the y-axis, then that's an even function.
00:09:27.000 --> 00:09:32.000
That's why I say it has mirror symmetry across the y-axis.
00:09:32.000 --> 00:09:35.000
That's what an even function looks like.
00:09:35.000 --> 00:09:38.000
There is a common misconception among students.
00:09:38.000 --> 00:09:43.000
People think, well with numbers, every number's either odd or even.
00:09:43.000 --> 00:09:50.000
People think, well, every function is odd or even and that's not necessarily true.
00:09:50.000 --> 00:10:00.000
Just for example, here's a line but that is not either
00:10:00.000 --> 00:10:07.000
that does not have a rotational symmetry around the origin nor does it have mirror symmetry around the y-axis.
00:10:07.000 --> 00:10:12.000
That function, this line, is not an add or an even function.
00:10:12.000 --> 00:10:16.000
It's a little misleading people think every function has to be an odd or an even function.
00:10:16.000 --> 00:10:18.000
That's not true.
00:10:18.000 --> 00:10:24.000
It's just true that some functions are odd, some functions are even, some functions are neither one.
00:10:24.000 --> 00:10:26.000
We'll practice some examples of that.
00:10:26.000 --> 00:10:35.000
First, we're going to look at some common angles and we're going to figure out what the cosines and sines are.
00:10:35.000 --> 00:10:46.000
Let me draw a big unit circle here.
00:10:46.000 --> 00:10:48.000
That's a circle of radius 1.
00:10:48.000 --> 00:10:55.000
Let's remember where these angles are, 0, of course is on the positive x-axis.
00:10:55.000 --> 00:11:00.000
Here's the x-axis, here's the y-axis, there is zero,
00:11:00.000 --> 00:11:06.000
π over 2, remember that's the same as a 90-degree angle, that's a right angle, so that's up there π over 2;
00:11:06.000 --> 00:11:11.000
π is over here, that's a 180 degrees;
00:11:11.000 --> 00:11:16.000
3π over 2, is down here, and 2π is right here.
00:11:16.000 --> 00:11:20.000
We want to find the cosine and sine of each one of those angles.
00:11:20.000 --> 00:11:29.000
Now remember, cosine and sine, by definition, are the x and y coordinates of those angles.
00:11:29.000 --> 00:11:31.000
What are these x and y coordinates?
00:11:31.000 --> 00:11:38.000
The 0 angle, it's x-coordinate is 1, and it's y-coordinate is 0.
00:11:38.000 --> 00:11:52.000
That tells us that cos(0) is 1, and that sin(0) is 0.
00:11:52.000 --> 00:12:06.000
π over 2 is up here, and so it's cosine is the x-coordinate,
00:12:06.000 --> 00:12:10.000
well, the x-coordinate of that point is 0.
00:12:10.000 --> 00:12:19.000
The y-coordinate is 1, and so that's the sin of π over 2.
00:12:19.000 --> 00:12:28.000
π is over here at (-1,0), so that's the cosine and sine of π.
00:12:28.000 --> 00:12:40.000
Cos(π) is -1, sin(π) is 0.
00:12:40.000 --> 00:12:52.000
Finally, 3π over 2 is down here at (0,-1), so that's the cosine and sine of 3π over 2.
00:12:52.000 --> 00:13:03.000
Cos(0), sin(π/2), is -1.
00:13:03.000 --> 00:13:10.000
And one more, 2π is back in the same place as 0, so it has the same cosine and sine.
00:13:10.000 --> 00:13:25.000
Cos(2π) is 1, sin(2π) is 0.
00:13:25.000 --> 00:13:28.000
That's how you figure out the cosines and sines of angles.
00:13:28.000 --> 00:13:33.000
As you graph them on the unit circle, and then you look at the x and y coordinates.
00:13:33.000 --> 00:13:39.000
The x-coordinate is always the cosine, and the y-coordinate is always the sine.
00:13:39.000 --> 00:13:46.000
By the way, these are very common values, 0, π/2, 3π/2, and 2π.
00:13:46.000 --> 00:13:50.000
You should really know the sines and cosines of these angles by heart.
00:13:50.000 --> 00:14:01.000
They come up so often in trigonometry context that it's worth memorizing these things, and being able to sort of regurgitate them very very quickly.
00:14:01.000 --> 00:14:09.000
If you ever forget them though, if you ever can't quite remember what the cosine of π/2, or the sine of 3π/2 is,
00:14:09.000 --> 00:14:17.000
Then, what you do is draw yourself a little unit circle, and you figure what the x and y coordinates are and you can always work them out.
00:14:17.000 --> 00:14:29.000
It's worth memorizing them to know them quickly, but if you ever get confused, you are not quite sure, just draw yourself a unit circle and you'll figure them out quickly.
00:14:29.000 --> 00:14:39.000
We're going to use these values, so I hope you will remember these values for the next example.
00:14:39.000 --> 00:14:48.000
In the next example, we're being asked to draw the graphs of the cosine and the sine functions, so let's remember what those values are.
00:14:48.000 --> 00:14:52.000
We're to label all the zeros, and the maxima and the minima of these functions.
00:14:52.000 --> 00:14:58.000
Let me set up some axis here.
00:14:58.000 --> 00:15:05.000
I'm going to label my x-axis in terms of multiples of π.
00:15:05.000 --> 00:15:10.000
The reason I'm going to do that is because we're talking about cosines and sines of multiples of π.
00:15:10.000 --> 00:15:15.000
We're really talking about radians here.
00:15:15.000 --> 00:15:29.000
That's π, that's 2π, that's π/2, and that's 3π/2.
00:15:29.000 --> 00:15:31.000
That's 0, of course.
00:15:31.000 --> 00:15:40.000
The y-axis, I'm going to label as 1 and -1.
00:15:40.000 --> 00:15:45.000
I've set up my scale here, remember that π is about 3.14 so it's a little bit bigger than 3.
00:15:45.000 --> 00:15:52.000
I've set up my scale here so that the π is about a little bit beyond 3 units on the graph.
00:15:52.000 --> 00:15:55.000
I'll extend it a little bit on the left here as well.
00:15:55.000 --> 00:16:03.000
We've got -π, I'll draw that around -3 and -π/2.
00:16:03.000 --> 00:16:08.000
I want to graph the sine and cosine function according to those values that we figured out.
00:16:08.000 --> 00:16:18.000
Remember that the sine and cosine function are correspond to the coordinates of angles on the unit circle.
00:16:18.000 --> 00:16:26.000
So sine and cosine,remember, are the x and y coordinates of angles on the unit circle.
00:16:26.000 --> 00:16:34.000
Now, those coordinates will never get bigger than 1 or smaller than -1.
00:16:34.000 --> 00:16:44.000
That's why on my y-axis, I only went up to -1 and 1, because the coordinates will never get bigger than -1 and 1.
00:16:44.000 --> 00:16:47.000
Let me start out with the cosine function.
00:16:47.000 --> 00:16:54.000
I'll do that one in blue, y=cos(x).
00:16:54.000 --> 00:16:58.000
We'll remember the values that we learned in the previous question.
00:16:58.000 --> 00:17:24.000
Cos(x), cos(0) is 1, cos(π/2) was 0, cos(π) is -1, cos(3π/2) is 0 and cos(2π) is 1.
00:17:24.000 --> 00:17:37.000
What you get is this smooth curve.
00:17:37.000 --> 00:17:45.000
After 2π, remember, after you circle 2π radians, then everything starts repeating.
00:17:45.000 --> 00:17:51.000
What happens after 2π is that it repeats itself.
00:17:51.000 --> 00:18:00.000
It repeats itself in the negative direction as well.
00:18:00.000 --> 00:18:05.000
Now we know what the graph of cos(x) looks like.
00:18:05.000 --> 00:18:12.000
I'll do the sine graph in red.
00:18:12.000 --> 00:18:30.000
Remember that sin(0) is 0, sin(π/2) is 1, sin(π) was 0 again, sin(3π/2) is -1, sin(2π) is 0.
00:18:30.000 --> 00:18:34.000
I'm doing this from memory and hopefully you remember these values as well.
00:18:34.000 --> 00:18:42.000
But if you can't remember these values, you know you can always look back at the unit circle and figure them out again just from their coordinates.
00:18:42.000 --> 00:18:59.000
The sine graph, I'm going to connect this up into a smooth curve.
00:18:59.000 --> 00:19:06.000
It repeats itself after this.
00:19:06.000 --> 00:19:11.000
It repeats in the negative direction as well.
00:19:11.000 --> 00:19:13.000
That's what y=sin(x) looks like.
00:19:13.000 --> 00:19:20.000
It actually has the same shape as cos(x) but it's shifted over on the graph.
00:19:20.000 --> 00:19:23.000
Now, we're asked to label all zeros, maxima and minima.
00:19:23.000 --> 00:19:27.000
Let me go through and label the zeros first.
00:19:27.000 --> 00:19:37.000
This is on the cos(x), this is (π/2,0), that's the 0 right there.
00:19:37.000 --> 00:19:42.000
There is one right there, (3π/2,0).
00:19:42.000 --> 00:19:57.000
This one, even though I haven't labeled it on the x-axis, is actually (5π/2,0), because it's π/2 beyond 2π,(-π/2,0).
00:19:57.000 --> 00:20:02.000
Those are the zeros of the cosine graph.
00:20:02.000 --> 00:20:07.000
The maxima, the high points, remember, cosine and sine never get bigger` than 1 or less than -1.
00:20:07.000 --> 00:20:13.000
Any time it actually hits 1, it's a maximum.
00:20:13.000 --> 00:20:18.000
They're the two maxima, at 0 and 2π.
00:20:18.000 --> 00:20:40.000
The minimum value is -1, so there is (π,-1) and the next one would be at (3π,-1), there is one at (-π,-1).
00:20:40.000 --> 00:20:45.000
Now, let me do the zeros of this sine graph.
00:20:45.000 --> 00:20:58.000
There is one (0,0), (-π,0), (π,0), and (2π,0).
00:20:58.000 --> 00:21:13.000
The maximum value would be 1 and that occurs at π/2, and again at 5π/2.
00:21:13.000 --> 00:21:18.000
The minimum value would be at 3π/2, where the sine is -1,
00:21:18.000 --> 00:21:22.000
Remember sine and cosine never go outside that range, -1 to 1,
00:21:22.000 --> 00:21:29.000
and at -π/2.
00:21:29.000 --> 00:21:41.000
All these values you should pretty much have memorized there the sort of simplest values, the easiest ones to figure out of sine and cosine.
00:21:41.000 --> 00:21:47.000
Let's try an example where we're using this trigonometric functions in a triangle.
00:21:47.000 --> 00:21:57.000
What we're told is that a right triangle has short sides of length 3 and 4.
00:21:57.000 --> 00:22:03.000
We're asked to find the sine, cosine, and tangent of all angles in the triangle.
00:22:03.000 --> 00:22:07.000
Remember, I haven't really explained what tangent is yet, but we did learn that formula SOH CAH TOA.
00:22:07.000 --> 00:22:10.000
That's what we're going to be using here.
00:22:10.000 --> 00:22:14.000
The first thing we need to figure out here is what the hypotenuse of this triangle is.
00:22:14.000 --> 00:22:29.000
We have the Pythagorean theorem that says, h² = 3² + 4², which is 9 plus 16, which is 25.
00:22:29.000 --> 00:22:34.000
That tells us that the hypotenuse must be 5.
00:22:34.000 --> 00:22:44.000
Now we're going to find the sine, cosine and tangent of each one of these angles.
00:22:44.000 --> 00:22:52.000
Let's figure out this angle first, so I'll call it θ, sin(θ), remember SOH CAH TOA,
00:22:52.000 --> 00:23:01.000
let me write that down for reference here, SOH CAH TOA,
00:23:01.000 --> 00:23:06.000
sin(θ) is equal to opposite over adjacent
00:23:06.000 --> 00:23:14.000
That's 4, that's the opposite side from θ over...
00:23:14.000 --> 00:23:21.000
Sorry, I said sin(θ) is opposite over adjacent, of course, sin(θ) is opposite over hypotenuse, and the hypotenuse is 5.
00:23:21.000 --> 00:23:24.000
So, sin(θ) there is 4/5.
00:23:24.000 --> 00:23:29.000
Cos(θ) is equal to adjacent over hypotenuse.
00:23:29.000 --> 00:23:36.000
Well, the side adjacent to θ is 3, hypotenuse is still 5.
00:23:36.000 --> 00:23:43.000
Tan(θ) is equal to opposite over adjacent.
00:23:43.000 --> 00:23:47.000
Again, we haven't really learned what a tangent means yet, but we can still use SOH CAH TOA.
00:23:47.000 --> 00:23:55.000
The opposite over adjacent is 4/3.
00:23:55.000 --> 00:24:02.000
Let me call the other angle here φ.
00:24:02.000 --> 00:24:11.000
Sin(φ) is equal to the opposite over the hypotenuse, so 3/5.
00:24:11.000 --> 00:24:23.000
Cos(φ) is equal to adjacent over hypotenuse, the adjacent angle beside φ is 4.
00:24:23.000 --> 00:24:32.000
Tan(θ) is equal to the opposite over adjacent, so that's 3/4.
00:24:32.000 --> 00:24:36.000
Finally, we have the right angle here, I'll call that α.
00:24:36.000 --> 00:24:40.000
We can't really use SOH CAH TOA on that, but I know that sin(α),
00:24:40.000 --> 00:24:49.000
α is a 90-degree angle, or in terms of radians, it's π/2.
00:24:49.000 --> 00:24:54.000
The sin(π/2), we learned before, is 1.
00:24:54.000 --> 00:25:05.000
Cos(α) is cos(π/2), and we learned that the cos(π/2) before was 0.
00:25:05.000 --> 00:25:14.000
Finally, tan(α) is tan(π/2), and we haven't really learned about tangent yet.
00:25:14.000 --> 00:25:18.000
In particular, we haven't learned what to do with the tan(π/2).
00:25:18.000 --> 00:25:23.000
But we'll get to that in a later lecture, and we'll learn that that's actually not defined.
00:25:23.000 --> 00:25:32.000
So we can't give a value to the tangent of π/2.
00:25:32.000 --> 00:25:41.000
All of these angles were things we worked out just using this one master formula, SOH CAH TOA.
00:25:41.000 --> 00:25:48.000
That tells you the sine, cosine and tangent of the small angles in the triangle.
00:25:48.000 --> 00:25:50.000
The SOH CAH TOA does not really apply to the right angle.
00:25:50.000 --> 00:25:58.000
But we already know the sine and cosine of a right angle, of a 90-degree angle, because we figured them out before.
00:25:58.000 --> 00:26:02.000
We'll try some more examples here.
00:26:02.000 --> 00:26:10.000
I want to try graphing the function sin(x + π/2) and cos(x - π/2).
00:26:10.000 --> 00:26:16.000
Then, I want to determine whether these functions are odd or even, or neither one.
00:26:16.000 --> 00:26:24.000
Well, something that's really good to remember here from your algebra class, or from the algebra lectures here on educator.com,
00:26:24.000 --> 00:26:33.000
is that you have a function, and you try to graph f(x) minus a constant,
00:26:33.000 --> 00:26:39.000
what that does is it moves the graph of the function over by the amount of the constant.
00:26:39.000 --> 00:26:42.000
That's very useful in the trigonometric setting.
00:26:42.000 --> 00:26:48.000
Let me start out by graphing f(x)=sin(x).
00:26:48.000 --> 00:26:55.000
And we did that in the previous example, and I remember what the sine graph looks like.
00:26:55.000 --> 00:27:15.000
It starts at 0, it peaks at π/2, it goes back to 0 at π, it bottoms out at 3π/2 at -1, and then it goes back to 0 at 2π.
00:27:15.000 --> 00:27:23.000
What I graphed there was just sin(x), I have not introduced this change yet.
00:27:23.000 --> 00:27:31.000
What I'm going to do, in blue now, is the sin(x + π/2).
00:27:31.000 --> 00:27:36.000
What that's going to do is going to move the graph over π/2 units.
00:27:36.000 --> 00:27:41.000
But remember there was a negative sign in there that I don't have here.
00:27:41.000 --> 00:27:52.000
This is really like, sin[x - (-π/2)].
00:27:52.000 --> 00:28:00.000
It moves the graph over -π/2 units, which means it moves it to the left π/2 units.
00:28:00.000 --> 00:28:11.000
I'm going to take this graph and I'm going to move it over to the left π/2 units.
00:28:11.000 --> 00:28:29.000
Now it's going to start at -π/2, come back down at π/2, bottom out at π, and come back to 0 at 3π/2.
00:28:29.000 --> 00:28:39.000
So there is -π/2, 0, π/2, and it comes back at 3π/2.
00:28:39.000 --> 00:28:45.000
That's what our graph of sin(x + π/2) looks like.
00:28:45.000 --> 00:28:50.000
Then the question is, is that odd or even, or neither?
00:28:50.000 --> 00:28:58.000
Well, remember there is a graphic way to look at the graph of a function in determining whether it's odd or even, or neither.
00:28:58.000 --> 00:29:10.000
An odd function, remember, has rotational symmetry, and even function has mirror symmetry across the y-axis.
00:29:10.000 --> 00:29:17.000
Well, if you look at this at this graph, it certainly does not have rotational symmetry.
00:29:17.000 --> 00:29:22.000
If you tried to rotate it around the origin, it would end up down here, and that would be a different graph.
00:29:22.000 --> 00:29:32.000
However, it does have mirror symmetry around the y-axis.
00:29:32.000 --> 00:29:41.000
So because it's mirror symmetric around the y-axis, sin(x + π/2).
00:29:41.000 --> 00:30:12.000
F(x) is an even function because it has mirror symmetry, mirror symmetry across the y-axis.
00:30:12.000 --> 00:30:16.000
OK, let us move on to the next one, cos(x - π/2).
00:30:16.000 --> 00:30:29.000
Again, I'm going to start with the basic cosine function that we learned how to graph in a previous example.
00:30:29.000 --> 00:30:41.000
So that my mark's here, there is π, there is 2π, π/2, 3π/2, 0, -π/2.
00:30:41.000 --> 00:31:02.000
Now, cosine had zero at 1, then it comes down to 0 at π/2, bottoms out at -1 at π, comes back to 0 at 3π/2, and by 2π, it's back up at 1.
00:31:02.000 --> 00:31:05.000
What I have just graphed there in black is cos(x).
00:31:05.000 --> 00:31:11.000
I have not tried to introduce the shift yet.
00:31:11.000 --> 00:31:18.000
But what we want to do is to graph cos(x - π/2).
00:31:18.000 --> 00:31:28.000
That's like saying, you see up here is π/2, that's going to shift the graph π/2 units to the right,
00:31:28.000 --> 00:31:30.000
because it's -π/2, it shifts it to the right.
00:31:30.000 --> 00:31:44.000
We take this graph and we move it over to the right π/2 units.
00:31:44.000 --> 00:31:50.000
I drew that a little too high, let me flatten that out a little bit.
00:31:50.000 --> 00:32:01.000
What we have there is the graph of cos(x - π/2), and of course that keeps going in the other direction there.
00:32:01.000 --> 00:32:10.000
We see, actually if you look carefully, cos(x - π/2) actually turns out be the same graph as sin(x).
00:32:10.000 --> 00:32:15.000
That's a familiar function if you remember those graphs.
00:32:15.000 --> 00:32:19.000
Again, we're being asked whether the function is odd or even, or neither.
00:32:19.000 --> 00:32:24.000
For odd, we're checking rotational symmetry around the origin.
00:32:24.000 --> 00:32:26.000
Look at that.
00:32:26.000 --> 00:32:33.000
If you rotate the graph 180 degrees around the origin, what you'll end up with is exactly the same picture.
00:32:33.000 --> 00:33:02.000
g(x) is an odd function because it has rotational symmetry around the origin.
00:33:02.000 --> 00:33:04.000
Is it an even function?
00:33:04.000 --> 00:33:07.000
Does it have mirror symmetry around the y-axis?
00:33:07.000 --> 00:33:16.000
No, it does not because it has this kind of bump on the right hand side, and it does not have the same bump on the left hand side, so it's not an odd function.
00:33:16.000 --> 00:33:18.000
Sorry, it's not an even function.
00:33:18.000 --> 00:33:20.000
It's just an odd function.
00:33:20.000 --> 00:33:24.000
What we did there was we started with the sine and cosine graphs that we remembered.
00:33:24.000 --> 00:33:28.000
it's worth memorizing the basic sine and cosine graphs.
00:33:28.000 --> 00:33:33.000
Then we examined the shift that each one introduced.
00:33:33.000 --> 00:33:36.000
Each one got shifted π/2 units to the right or left.
00:33:36.000 --> 00:33:39.000
Then we drew the new graphs.
00:33:39.000 --> 00:33:42.000
Then we looked back at them and we checked what kind of symmetry do they have.
00:33:42.000 --> 00:33:47.000
Do they have rotational symmetry or mirror symmetry?
00:33:47.000 --> 00:43:16.000
And that tells us whether they are odd or even.