WEBVTT mathematics/trigonometry/murray
00:00:00.000 --> 00:00:03.000
Hi, these are the trigonometry lectures on educator.com.
00:00:03.000 --> 00:00:09.000
Today we're going to learn about sine and cosine values of special angles.
00:00:09.000 --> 00:00:14.000
When I say these special angles, there are certain angles that you really want to know by heart.
00:00:14.000 --> 00:00:22.000
Those are the 45-45-90 triangle, and the 30-60-90 triangle.
00:00:22.000 --> 00:00:28.000
Let me talk about the 45-45-90 triangle first.
00:00:28.000 --> 00:00:41.000
I'll draw this in blue.
00:00:41.000 --> 00:00:52.000
Here's a 45-45-90 triangle and I'm going to say that each side has length 1.
00:00:52.000 --> 00:01:06.000
If each of the short sides has length 1, by the Pythagorean theorem, we can figure out that the long side, the hypotenuse, would have length square root of 2.
00:01:06.000 --> 00:01:11.000
I'm going to scale this triangle down a little bit now.
00:01:11.000 --> 00:01:15.000
I wanted to scale it down so the hypotenuse has length 1.
00:01:15.000 --> 00:01:19.000
That means I have to divide all three sides by square root of 2.
00:01:19.000 --> 00:01:35.000
If I scale this down, so the hypotenuse has length 1 that means the shorter sides has length 1 over the square root of 2, because I divided each side by square root of 2.
00:01:35.000 --> 00:01:42.000
Then if you rationalize that, the way you learned in your algebra class, multiply top and bottom by square root of 2.
00:01:42.000 --> 00:01:53.000
You get square root of 2 over 2, and square root of 2 over 2.
00:01:53.000 --> 00:02:02.000
Those are very important values to remember because those are going to come up as sines and cosines of our 45-degree angles on the next slide.
00:02:02.000 --> 00:02:09.000
First, I'd like to look also at the 30-60-90 triangle.
00:02:09.000 --> 00:02:13.000
I have to do a little geometry to work this out for you.
00:02:13.000 --> 00:02:22.000
I'm going to start with an equilateral triangle, a triangle where all three sides have 60 degrees.
00:02:22.000 --> 00:02:25.000
I'm going to have assume that each side has length 2.
00:02:25.000 --> 00:02:30.000
The reason I'm going to do that is because I'm going to divide that triangle in half.
00:02:30.000 --> 00:02:40.000
If we divide that triangle in half, then we get a right angle here and each one of these pieces will have length 1.
00:02:40.000 --> 00:02:49.000
Now, if I just look at the right hand triangle.
00:02:49.000 --> 00:02:55.000
Remember that each one of the corners of the original triangle was 60 degrees.
00:02:55.000 --> 00:03:02.000
That means that the small corner is 30 degrees, and I have a right angle here.
00:03:02.000 --> 00:03:06.000
Now, the short side has length 1, the long side has length 2.
00:03:06.000 --> 00:03:11.000
I'm going to figure out what the other side is using the Pythagorean theorem.
00:03:11.000 --> 00:03:14.000
Let me call that x for now.
00:03:14.000 --> 00:03:22.000
I know that x² + 1² = 2², which is 4.
00:03:22.000 --> 00:03:29.000
So, x² = 4 - 1, which is 3.
00:03:29.000 --> 00:03:33.000
So, x is the square root of 3.
00:03:33.000 --> 00:03:39.000
That's where I got this relationship, 1, square root of 3, 2.
00:03:39.000 --> 00:03:43.000
There's the 1, there's the square root of 3, and there's the 2.
00:03:43.000 --> 00:03:52.000
Now, I'm going to turn this triangle on its side, and I want to scale it down.
00:03:52.000 --> 00:03:58.000
Originally, it was 1, square root of 3, 2.
00:03:58.000 --> 00:04:04.000
But again, I want to scale the triangle down so that the hypotenuse has length 1.
00:04:04.000 --> 00:04:07.000
To do that, I have to divide everything by 2.
00:04:07.000 --> 00:04:11.000
So the short side now has length 1/2.
00:04:11.000 --> 00:04:18.000
The longer of the two short sides has length square root of 3 over 2.
00:04:18.000 --> 00:04:22.000
Remember that's the side adjacent to the 30-degree angle.
00:04:22.000 --> 00:04:24.000
That's the side adjacent to the 60-degree angle,
00:04:24.000 --> 00:04:27.000
That's the right hand side.
00:04:27.000 --> 00:04:33.000
These two triangles are very key to remember, in remembering all the sines and cosines.
00:04:33.000 --> 00:04:42.000
In fact, if you can remember these lengths of these two triangles, you can work out everything else just from these two triangles.
00:04:42.000 --> 00:04:57.000
Let me emphasize again, the 45-45-90 triangle, its sides have length 1, square root of 2 over 2, and square root of 2 over 2.
00:04:57.000 --> 00:05:10.000
The 30-60-90 triangle, has sides of length 1, 1/2, and root 3 over 2.
00:05:10.000 --> 00:05:12.000
Those are the values that you need to remember.
00:05:12.000 --> 00:05:21.000
If you can remember those, you can work out all the sines and cosines you need to know for every trigonometry class ever.
00:05:21.000 --> 00:05:24.000
Let's explore those a little bit.
00:05:24.000 --> 00:05:28.000
We already figured out, let me draw a unit circle.
00:05:28.000 --> 00:05:38.000
We know that sines and cosines occur as the x and y coordinates of different angles.
00:05:38.000 --> 00:05:41.000
Well, if you're at 0...
00:05:41.000 --> 00:05:59.000
Let me just draw in some key angles here, 0, here's 90, here's 45, here's 30, and here's 60.
00:05:59.000 --> 00:06:08.000
If you're at 0 degrees, which is the same as 0 radians, then the cosine and sine, the x and y coordinates are just 1 and 0.
00:06:08.000 --> 00:06:10.000
We already figured those out before.
00:06:10.000 --> 00:06:14.000
The other easy one is the 90-degree angle up here.
00:06:14.000 --> 00:06:20.000
We figured out that that's π/2 radians, and the cosine and sine are 0, and 1 there.
00:06:20.000 --> 00:06:26.000
Now, the new ones, let me start with 45, because I think that one's a little bit easier.
00:06:26.000 --> 00:06:31.000
The 45-degree angle, there it is right there.
00:06:31.000 --> 00:06:37.000
We want to figure out what the x and y coordinates are because those give us the sine and cosine.
00:06:37.000 --> 00:06:47.000
Well, we just figured out that a triangle that has 1 as its hypotenuse has square root of 2 over 2, as both its x and y sides.
00:06:47.000 --> 00:06:58.000
That's where we get the square root of 2 over 2 as the cosine and sine of the 45-degree angle, also known as π/4 radians.
00:06:58.000 --> 00:07:02.000
For the 30-degree angle, I'll do this one in blue.
00:07:02.000 --> 00:07:11.000
The 30-degree angle, we have again, hypotenuse has length 1.
00:07:11.000 --> 00:07:16.000
Remember, the length of the long side is root 3 over 2.
00:07:16.000 --> 00:07:19.000
And the length of the short side is 1/2.
00:07:19.000 --> 00:07:27.000
That's how you know the sine and cosine of the 30-degree angle or π/6.
00:07:27.000 --> 00:07:32.000
The cosine, the x-coordinate, root 3 over 2, sine is 1/2.
00:07:32.000 --> 00:07:48.000
The 60-degree angle, that's just the same triangle but it's flipped the other way so that the long side is on the vertical part and the short side is on the horizontal axis.
00:07:48.000 --> 00:07:53.000
The short side is 1/2.
00:07:53.000 --> 00:07:58.000
The long side is now the y-axis, that's root 3 over 2.
00:07:58.000 --> 00:08:08.000
That's how we get 1/2 being the cosine of 60 degrees, root 3 over 2 being the sine of 60 degrees.
00:08:08.000 --> 00:08:14.000
These values are really worth memorizing but you remember that you figure all out from those two triangles.
00:08:14.000 --> 00:08:29.000
All you need to know is that one triangle has length 1, has hypotenuse 1, and sides root 2 over 2, that's the 45-45-90 triangle.
00:08:29.000 --> 00:08:46.000
The other triangle has hypotenuse 1, and then the long side is root 3 over 2, short side is 1/2, that's the 30-60-90 triangle.
00:08:46.000 --> 00:08:54.000
Just take that triangle and you flip it whichever way you need to, to get the angle that you're looking for.
00:08:54.000 --> 00:09:06.000
These angles, these sines and cosines are the key ones to remember, root 2 over 2, root 3 over 2, and 1/2.
00:09:06.000 --> 00:09:33.000
From that, what you're going to do is figure out the sines and cosines of all the other angles all over the unit circle.
00:09:33.000 --> 00:09:35.000
Here's my unit circle.
00:09:35.000 --> 00:09:43.000
We figured out all the sines and cosines of all the angles in the first quadrant.
00:09:43.000 --> 00:09:49.000
All we have to do now is figure out all the sines and cosines of the angles in the other quadrants.
00:09:49.000 --> 00:09:50.000
Let me draw them out.
00:09:50.000 --> 00:09:53.000
But it's the same numbers every time.
00:09:53.000 --> 00:10:01.000
All you have to do is figure out whether those numbers are positive or negative, and that just depends on which quadrant you're in.
00:10:01.000 --> 00:10:07.000
All you have to do is remember those key numbers, root 2 over 2, root 3 over 2 and 1/2.
00:10:07.000 --> 00:10:13.000
Then you're going to figure out which ones are positive in which quadrants.
00:10:13.000 --> 00:10:22.000
Let me show you how you'll remember that, 1, 2, 3, 4.
00:10:22.000 --> 00:10:30.000
Remember that the sine is the y value, and the cosine is the x value.
00:10:30.000 --> 00:10:36.000
In the first quadrant, both the x's and y's are positive, and so sines and cosines are going to positive.
00:10:36.000 --> 00:10:43.000
I'm going to write that as an a, which stands for all the values of everything is positive.
00:10:43.000 --> 00:10:53.000
In the second quadrant, over here, the x values are negative, the y values are positive.
00:10:53.000 --> 00:10:58.000
Now, the x values correspond to the cosines, the cosines are negative and the sines are positive.
00:10:58.000 --> 00:11:07.000
I'm going to write as s here, for the sines being positive.
00:11:07.000 --> 00:11:09.000
In the third quadrant, both x's and y's are negative.
00:11:09.000 --> 00:11:15.000
X and y are both negative, that means cosines and sines are both negative.
00:11:15.000 --> 00:11:23.000
Tangent, we haven't learned the details of tangent yet but we're going to learn later that tangent is sine over cosine.
00:11:23.000 --> 00:11:27.000
Both sine and cosine are negative, that means sine over cosine is positive.
00:11:27.000 --> 00:11:31.000
It turns out that tangent is positive down that third quadrant.
00:11:31.000 --> 00:11:33.000
We'll learn the details of tangent later.
00:11:33.000 --> 00:11:38.000
In the meantime, we'll just remember that tangent is positive in the third quadrant.
00:11:38.000 --> 00:11:45.000
In the fourth quadrant here, that was the third quadrant that we just talked about, now we're moving on to the fourth quadrant.
00:11:45.000 --> 00:11:48.000
The x's are positive now, the y's are negative.
00:11:48.000 --> 00:11:53.000
That means the cosine is positive, but the sine is negative.
00:11:53.000 --> 00:11:58.000
I'll list the cosine here because I'm listing the positive ones.
00:11:58.000 --> 00:12:00.000
First quadrant, they're all positive.
00:12:00.000 --> 00:12:02.000
Second quadrant, sines are positive.
00:12:02.000 --> 00:12:04.000
Third quadrant, tangents are positive.
00:12:04.000 --> 00:12:07.000
Fourth quadrant, cosines are positive.
00:12:07.000 --> 00:12:13.000
The way you remember that is with this little acronym, All Students Take Calculus.
00:12:13.000 --> 00:12:21.000
That shows you as you go around the four quadrants, All Students Take Calculus.
00:12:21.000 --> 00:12:24.000
It shows you which ones are positive in each quadrant.
00:12:24.000 --> 00:12:26.000
First quadrant are all positive.
00:12:26.000 --> 00:12:28.000
Second quadrant sines are positive.
00:12:28.000 --> 00:12:30.000
Third quadrant, tangents are positive.
00:12:30.000 --> 00:12:33.000
Fourth quadrant, cosines are positive.
00:12:33.000 --> 00:12:36.000
That's how you'll remember what the signs are in each quadrant.
00:12:36.000 --> 00:12:38.000
That means the positive signs and the negative signs in each quadrant.
00:12:38.000 --> 00:12:48.000
The numbers are just all these values that you've just memorized, root 3 over 2, root 2 over 2, and 1/2.
00:12:48.000 --> 00:12:55.000
We'll do some practice finding sines and cosines of values in other quadrants, based on this table that we remember.
00:12:55.000 --> 00:13:00.000
And these common values in common triangles that we remember.
00:13:00.000 --> 00:13:07.000
Then we'll take those values, introduce some positive and negative signs, and we'll come up with the sines and cosines of angles in other quadrants.
00:13:07.000 --> 00:13:11.000
Let's try some examples.
00:13:11.000 --> 00:13:20.000
First example here is a 120 degrees, you want to convert that to radians, identify it's quadrant, and find it's cosine and sine.
00:13:20.000 --> 00:13:26.000
First things first, let's convert it to radians, 120 times π/180 is 2π/3, because 120/180 is 2/3 so that's 2π/3 radians.
00:13:26.000 --> 00:13:34.000
Identify it's quadrant.
00:13:34.000 --> 00:13:40.000
Well, let me graph out my unit circle here.
00:13:40.000 --> 00:14:00.000
There's 0, there's π/2, π, 3π/2, and then 2π.
00:14:00.000 --> 00:14:05.000
Now, 2π/3 is between π/2 and π.
00:14:05.000 --> 00:14:08.000
In fact, it's closer to π/2.
00:14:08.000 --> 00:14:12.000
It's 2/3 of the way around from 0 to π.
00:14:12.000 --> 00:14:22.000
If you like that in terms of degrees, π is 180 degrees, and so 2π/3 is 2/3 of the way over to 180.
00:14:22.000 --> 00:14:27.000
Now, we're going to find cosine and sine.
00:14:27.000 --> 00:14:28.000
Let me show you how to do this.
00:14:28.000 --> 00:14:30.000
You draw a triangle here.
00:14:30.000 --> 00:14:33.000
Remember, we're looking for the x and y coordinates.
00:14:33.000 --> 00:14:35.000
Draw a triangle here.
00:14:35.000 --> 00:14:37.000
That's a 30-60 triangle.
00:14:37.000 --> 00:14:47.000
This is 60, that's a 60-degree angle.
00:14:47.000 --> 00:14:51.000
That's a 30-degree angle.
00:14:51.000 --> 00:14:57.000
We know what the lengths of these different sides are.
00:14:57.000 --> 00:15:07.000
We know that the long side there is root 3 over 2 and the short side there is 1/2.
00:15:07.000 --> 00:15:10.000
We know, remember, the cosines and sine are the x and y coordinates.
00:15:10.000 --> 00:15:30.000
The cosine of 120 or 2π/3 is 1/2, except that we're going to have to check whether that's positive or negative.
00:15:30.000 --> 00:15:35.000
Remember All Students Take Calculus.
00:15:35.000 --> 00:15:42.000
In the second quadrant, only the sine is positive, so the cosine must be negative.
00:15:42.000 --> 00:15:50.000
The sine of 2π/3, the y value is root 3 over 2.
00:15:50.000 --> 00:15:54.000
In the second quadrant, sines are positive, so that's positive.
00:15:54.000 --> 00:15:58.000
Our cosine and sine are -1/2 and root 3 over 2.
00:15:58.000 --> 00:16:04.000
If you didn't remember the All Students Take Calculus thing, you can also just work it out once you know what quadrant it's in.
00:16:04.000 --> 00:16:14.000
It's in quadrant 2 and we know there that the x coordinates are negative, and the y coordinates are positive.
00:16:14.000 --> 00:16:21.000
The cosine must be negative and the sine must be positive.
00:16:21.000 --> 00:16:31.000
The whole point of this is that you only really need to memorize the values of the triangles, root 2 over 2, root 3 over 2 and 1/2.
00:16:31.000 --> 00:16:45.000
Once you know those basic triangles, you can work out what the sines and cosines are in any different quadrant just by drawing in those triangles and then figuring out which ones have to be positive, and which ones are to be negative.
00:16:45.000 --> 00:16:49.000
Let's try another one.
00:16:49.000 --> 00:16:58.000
This one is converting 5π/3 radians to degrees, identifying it's quadrant, and finding its cosine and sine.
00:16:58.000 --> 00:17:20.000
5π/3 times 180/π, the π's cancel, and the we have 5/3 of 180, 180 over 3 is 80, so this is 5 times 60 is 300 degrees.
00:17:20.000 --> 00:17:33.000
Let's try and find that in the unit circle.
00:17:33.000 --> 00:17:54.000
We have 0, π/2 which is 90, π which is 180, 3π/2 which is 270, and 2π which is the same as 360 degrees.
00:17:54.000 --> 00:18:00.000
Now, 5π/3, that's bigger than π and that's smaller than 2π.
00:18:00.000 --> 00:18:10.000
In fact, that's π + 2π/3.
00:18:10.000 --> 00:18:19.000
That's π, which is right here, plus 2π/3.
00:18:19.000 --> 00:18:23.000
So, 2/3 the way around from π to 2π.
00:18:23.000 --> 00:18:24.000
There it is right there.
00:18:24.000 --> 00:18:28.000
That's in the fourth quadrant.
00:18:28.000 --> 00:18:30.000
So, we figured out what quadrant it's in.
00:18:30.000 --> 00:18:41.000
If you like degrees better, 300 degrees is a little bigger than 270, in fact, it's 30 degrees past 270, and 60 degrees short of 360.
00:18:41.000 --> 00:18:46.000
That's how you know that that angle is in that quadrant.
00:18:46.000 --> 00:18:48.000
Now we have to find its cosine and sine.
00:18:48.000 --> 00:18:54.000
That's the x and y coordinates.
00:18:54.000 --> 00:18:57.000
We set up our triangle there.
00:18:57.000 --> 00:19:02.000
We already know that that's a 60-degree angle, because it's 60 degrees short of 360.
00:19:02.000 --> 00:19:07.000
That's a 30-degree angle and we just remember our common values.
00:19:07.000 --> 00:19:14.000
The horizontal value, that's the short one, that's 1/2, that's the long one, root 3 over 2.
00:19:14.000 --> 00:19:22.000
We know our values are going to be 1/2 and root 3 over 2, we'll just have to figure out which one's positive and which one's negative.
00:19:22.000 --> 00:19:31.000
I know my cosine of 5π/3 is going to be either positive or negative 1/2.
00:19:31.000 --> 00:19:39.000
The sine of 5π/3 is positive or negative root 3/2.
00:19:39.000 --> 00:19:44.000
Remember All Students Take Calculus.
00:19:44.000 --> 00:19:49.000
Down there in the fourth quadrant, the cosine is positive and the sine is negative.
00:19:49.000 --> 00:19:59.000
If you don't remember All Students Take Calculus, you just look that you're in the fourth quadrant, x coordinates are positive, y coordinates are negative.
00:19:59.000 --> 00:20:02.000
You know which one's positive or negative.
00:20:02.000 --> 00:20:09.000
All you have to remember are those key values, 1/2, root 3 over 2, root 2 over 2.
00:20:09.000 --> 00:20:12.000
Remember those key values for the key triangles.
00:20:12.000 --> 00:20:19.000
Then it's just a matter of drawing the right triangle in the right place and figuring out which one is positive, and which one is negative.
00:20:19.000 --> 00:20:23.000
Let's try another one.
00:20:23.000 --> 00:20:24.000
This one is kind of tricky.
00:20:24.000 --> 00:20:29.000
This one's going to be challenging us to go backwards from the sine.
00:20:29.000 --> 00:20:42.000
We have to find all angles between 0 and 2π whose sine is -1/2.
00:20:42.000 --> 00:20:49.000
This is kind of foreshadowing the arc sine function that we'll be studying later on and one of the later lectures.
00:20:49.000 --> 00:20:52.000
In the meantime, the sine is -1/2.
00:20:52.000 --> 00:20:56.000
Remember now, the sine is the y-coordinate.
00:20:56.000 --> 00:20:59.000
We want things whose y coordinates are -1/2.
00:20:59.000 --> 00:21:06.000
I'm going to draw -1/2 on the y-axis, -1/2.
00:21:06.000 --> 00:21:12.000
I'm going to look for all angles whose y-coordinate is -1/2.
00:21:12.000 --> 00:21:16.000
Look, there's one right there.
00:21:16.000 --> 00:21:17.000
And there's one right there.
00:21:17.000 --> 00:21:20.000
I'm going to draw those in.
00:21:20.000 --> 00:21:29.000
I'm going to draw those triangles in.
00:21:29.000 --> 00:21:38.000
I know now that if we have a vertical component of 1/2, the horizontal component has to be root 3 over 2.
00:21:38.000 --> 00:21:47.000
That's because we remember those common triangles, 1/2, root 3 over 2, root 2 over 2.
00:21:47.000 --> 00:21:52.000
We're going to figure out what those angles are.
00:21:52.000 --> 00:21:58.000
I know that's a 30-degree angle.
00:21:58.000 --> 00:22:02.000
I know that that is 180.
00:22:02.000 --> 00:22:10.000
The whole thing is 210 degrees.
00:22:10.000 --> 00:22:14.000
I know that that is 30.
00:22:14.000 --> 00:22:17.000
I know that that must be 60.
00:22:17.000 --> 00:22:22.000
Remember, this is 270 degrees down here.
00:22:22.000 --> 00:22:25.000
We have 270 degrees plus 60 degrees.
00:22:25.000 --> 00:22:37.000
This is getting a little messy, so I'm going to redraw it over here.
00:22:37.000 --> 00:22:40.000
That's the angle we're trying to chase down here.
00:22:40.000 --> 00:22:42.000
We know that's 60.
00:22:42.000 --> 00:22:46.000
That much is 270.
00:22:46.000 --> 00:22:52.000
So, 270 plus 60 is 330 degrees.
00:22:52.000 --> 00:22:58.000
Those are the two angles that we're after, 210 degrees, 330 degrees.
00:22:58.000 --> 00:23:02.000
Let me convert those into radians.
00:23:02.000 --> 00:23:10.000
If you multiply that by π/180 then that's equal to...
00:23:10.000 --> 00:23:17.000
Let's see, that's 7π/6 radians.
00:23:17.000 --> 00:23:29.000
This one times π/180 is equal to 11π/6 radians.
00:23:29.000 --> 00:23:33.000
We've got our two angles in degrees and radians.
00:23:33.000 --> 00:23:40.000
The quadrants, the first one was in the third quadrant, quadrant 3.
00:23:40.000 --> 00:23:46.000
The second one was in the fourth quadrant, quadrant 4.
00:23:46.000 --> 00:23:56.000
Those are the two angles in both degrees and radians that had a sign of -1/2.
00:23:56.000 --> 00:24:02.000
Their y value was -1/2.
00:24:02.000 --> 00:24:14.000
What this comes down to is knowing those common values, 1/2 root 3 over 2, root 2 over 2.
00:24:14.000 --> 00:24:21.000
Once you know those common values, it's a matter of looking at the different quadrants and figuring out whether the x and y values are positive or negative.
00:24:21.000 --> 00:24:28.000
In this case, we had the sine, sine remember is the y-coordinate.
00:24:28.000 --> 00:24:33.000
Since it was negative, we knew that we had to be at -1/2 on the y-axis.
00:24:33.000 --> 00:24:44.000
We found -1/2 on the y-axis, drew in the triangles, recognized the 30-60 triangles that we've been practicing and then we were able to work out the angles.
00:24:44.000 --> 00:33:05.000
We'll try some more examples of that later.