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0:00
Hey, welcome back to Number Buddy. My name is Buddy. This is episode two in our playlist about trigonometry. So we're going to learn what to do instead of memorizing the Unit circle. So if you hear the word unit circle and you feel an inner sense of dread, I'm with you. Let's get into it. We're going to learn what to do. So it's not so much that the unit circle contains incorrect information. It's not wrong. It's just that when you memorize the unit circle.
0:27
You get away from dealing with right triangles as much, and the whole basis of trigonometry is right triangles, so we want to stick to doing everything in the mindset and keeping in mind that really what we're working with is a right triangle. Let's look at what the unit circle is. first of all,
0:48
you may have seen this before. I may have drawn it a little bit different than what you're what you're told. So we're told we have a circle of radius one. And we want to know what are these? X y coordinates. Of everywhere around the circle, or at least at certain angle increments at zero, at 30 degrees, at 45 degrees, at 60 degrees. You want to figure out what is the X value of this coordinate here that intersects the unit circle at 30 degrees. So for example, that's this one I've drawn here.
1:26
What is this x value here and what is the Y value? But there's never really a scenario that you need to know off the top of your head. What is the the X and Y coordinate for 135 degrees on the unit circle? It's just an artificial situation. Here's what I want you to do instead. So what we're going to do is learn a couple.
1:52
Of special right triangles. We're going to draw a special red triangle. Whose angle above the x axis is 30 degrees? We learned in the last video what we're going to do is take the end of that angle or at least the end of the segment. We draw representing that angle and draw it straight down to the x axis.
2:14
Lastly, I'm going to draw in. Aside here, so that we have an you that represents this horizontal, so a y value represents the vertical side, and then our hypotenuse is going to be Z or C. Rather than memorizing the coordinates of an x y point that intersects a circle of radius one at an angle of 30 degrees. Instead, we're just going to learn about the ratios of any 30, 6090 triangle.
2:48
So when we drew this line down here, we formed a right angle there because this angle here is 30 degrees above the x axis. We know this is 60 degrees. OK, we've learned that the hypotenuse is always across from the 90 degree angle.
3:07
Well, another similar related idea is that the shortest side of your right triangle is always across from the smallest angle. Starting with that side, the ratios of a 30, 6090 triangle are always one.
3:23
To Route three.
3:27
To to.
3:29
In other words, the side across from your 30 degree angle is what we're going to call our base unit.
3:37
And whatever that side is, the hypotenuse is always going to be twice that. So say, for example, if this side was. six.
3:49
OK, if this side was six, our hypotenuse would have to be twice that. So our hypotenuse would be twelve. And our other side, our.
4:01
Longer than this side, but still not the hypotenuse side that is horizontal right now is going to be six times Route three. So whatever that base unit is, which is across from the 30 degree angle is going to be what you start with. To get the other non hypotenuse side, you're going to take that and multiply it by the square root of three.
4:23
To get the hypotenuse side, you're going to take that shorter side and just double it. This works for any 30, 6090 triangle. No matter what you start with, you could start with 100.
4:34
This side would still be 103. And this side would still be 200.
4:41
OK, so it's the ratio that's helpful, not memorizing the coordinates of the intersection with a radius one circle.
4:50
When we get into actual trig functions like sine cosine tangent, so on.
4:56
We're going to learn that trig functions are just ratios, trig functions are representative of ratios, so we always want to be dealing in ratios of sides of right trials. So let's put this back on there, just in its simplest form. And we're going to say this is one. This is one times Route three. Or you can just ride it as three. And this is, too, you can write this just as Route three, as long as you remember, the reason we got three is we multiplied this shorter side by Route three.
5:31
It's not just inherently Route three. OK.
5:35
Very nice. So let's say instead, what if I gave you this information, what if I said, we're going to start with a Z or a hypotenuse side? We have a 30, 60, 90 triangle and we have a Z, that is. 22.
5:53
OK. Well, the first thing I might do is I might come down here.
5:59
And redraw a 30, 60, 90 triangle just loosely. Right. And I might draw my proportions, I know this is one, I know this is three, I know this is two. OK, so if I'm given that Z is 22 or my hypotenuse is two, I know that's twice of what the shortest side is. So if 22 is twice something, that something is eleven. So the shortest side is eleven. Now I have my base unit. And to get to this other side, this horizontal side, right now, we need to take eleven and multiply it by.
6:37
Route three. And we've just solved our right triangle. We could check this if we wanted to into a squared plus.
6:47
B squared equals. C squared, right? And this one is eleven three. Squared.
6:58
Plus, eleven squared. Equals.
7:03
What let's just say equals what so we can check it. OK. eleven squared is one. 21 times square root of three squared is just three.
7:13
Plus, eleven squared is one, 21. Very nice. 1:21 times, three is going to be three, 63.
7:25
Plus, one, 21, you may have also been able to see that this is just like multiplying 1:21 times for.
7:31
But regardless, we're going to end up with four 84 for 84 equals.
7:40
See Square. If we come over here and take 22 times 20 to two times two is four, two times two is four, two times two is four, two times two is 44 84.
7:55
So 22 square.
7:59
Is for four. And this checks out, so I said this ratio of sides works for any 30, 60, 90 triangle, which means it doesn't just have to be a triangle oriented like this. So we can imagine what if we took this and flipped it facing this direction?
8:21
This is still a 30, 60, 90 trial, my handwriting is all backwards. But we can understand that the same sides are there. This side is one. This side is Route three and this side is too right. I know this side has one because it's across from the smaller angle.
8:41
But now, if this is a 30 degree above this 180 degree, right, this represents an angle.
8:50
Of 150. So rather than memorizing both 30 degrees and 150 degrees, now we can just think of these in terms of ratios. The only difference is here because we have a triangle facing this direction. Like we said in our first video,
9:09
this side is going to be negative for three because we're measuring. Our side as an ex value that goes from the origin.
9:19
To that point there. The X value the side terminates, that is negative route three. OK. You can check the Pythagorean theorem, even though it's negative squaring it will still wind up with the same situation as a matter of fact. Any increment of 30 degrees, with the exception of your 90 degrees and your 180 80 degrees and so on. With the exception of your of your axes themselves, what have we said? This is a 60 degree angle. Well, we would still draw our line down this way.
9:56
And we would still make our third side. And we would see that our right angle is formed right here.
10:04
And we would know that if this was 60 degrees, the site was 30. Now the short side, instead of being vertical, the short side is flipped and the short side is horizontal. And this is a short side, which means this side has a ratio of three. And this side is to. So that, in other words, if this instead of being one, let's say, if this side was five.
10:29
This horizontal side was five, that means our vertical side would be five or three.
10:35
And our hypotenuse would be ten, and it works the same way all around the X y plane.
10:41
Any angle that is an increment of 30 degrees, 30 degrees, 60 degrees, 120 degrees. 150 degrees. 210 degrees C 240 degrees. 300 degrees. 330 degrees. These all correspond to special right triangles. 30, 6090 triangles,
11:08
right. We can draw these like that would look like that or that would look like that. This one would look like this or that one would look like that. And this one would look like that and so on.
11:22
We still have special right triangles. This is an exactly to scale the technically these vertical lines should line up, but we can form these right triangles out of them. The only thing to keep in mind is where your X values and your Y values are going to be negative. For instance, if we were talking about this 240 degree angle down here,
11:43
this would be a 60 degree angle. This would be a 30 degree angle, which means this side would be our short side because it's facing in the negative x direction. It would have a ratio of negative one. This other side, that's our long side, but not the hypotenuse would have a ratio of three. But because it's pointing down, it's going to be negative at three.
12:02
Hypotenuse is always positive. Other sides can be negative or positive, just depending on what direction your special right triangle is facing. OK, so we've talked about a 30, 6090 degree triangle. The other special right triangle to think about is a 45, 4590 triangle. Let's look at those. So from the x axis.
12:25
We're going 45 degrees sweeping this way.
12:30
When we do that.
12:33
We then take the end of this drought straight down and form our right angle. OK, reformer writing there. And now we have two sides here and hypotenuse.
12:45
Well, if this angle is 45 degrees.
12:51
This other angle also has to be 45 degrees, because 45 plus 45 is 90, plus our right angle gives us 180. So if we talked about how in a 30, 60, 90 triangle, the short side is across from the 30 degree angle and the longer but not hypotenuse side is across from the 60. What happens if both of our angles are the same? And the answer is they are the same. Our ratios of sides, whatever these two sides are, these two sides ratios are going to be one to one.
13:23
Whatever this side is, this side is also going to be the same. OK. So when I say one to one, I mean, it does. They don't have to be one. This side could be ten, but that would mean this side also has to be ten. That's a ratio of one to one. So we're to leave just the ratios up for a moment. When when we have one to one, let's see what happens with the hypotenuse, right, we can figure this out with.
13:50
The Pythagorean theorem, a squared plus b squared equals C squared. Well, if we have one squared
13:58
plus one squared, what does that equal? one squared is one. This other one squared is also one is just going to be two.
14:09
So, too is going to have to equal.
14:13
Our C squared. Which means when we take the square root of both sides, see is going to be positive or negative square root of two. But because we're dealing with hypotenuse. We're only going to take the positive one. So whenever you have a 45, 45, 90 special right triangle, our ratios of our sides are going to be one to one to square root of two. And that's these here. The sides always have to be the same. And that's going to make this square root of two times that. So for example, if this side.
14:48
Was ten. This side would also have to be ten, which would mean the hypotenuse is ten times the square root of two. OK. It's not just the square root of two. It's whatever your base unit is rich in these, in this case, is the same times that square root of two.
15:06
Let's look at what a 45, 45, 90 triangle looks like in a different quadrant. If we go this way, this would have been an angle of.
15:15
135, an angle of 135 from the x axis leaves us 45 to get down to the hundred and 80 degree mark.
15:27
So now we have a similar setup.
15:32
But we're facing this way. So we have 45. We have 90. And if this is 45, we know that's 45, if we wanted to just keep our simple ratios and say one to one to Route two. We have 11 route too. Again, the only difference is since our triangle is facing this direction, the negative x direction.
15:55
This side has a negative side ratio. OK.
16:01
Same thing works. Down in this direction.
16:07
If we have 45 degrees this way. We have a triangle that looks like this.
16:15
We form a right angle there, right? Our ratios of our sides, we can actually keep this in the same idea here, so this is still a negative one on this side because our ex is still facing in the negative direction.
16:28
The only difference is now are why is facing in the negative direction to.
16:34
This is starting to come together. Lastly, let's go ahead and finish this up and look at. Look at our fourth quadrant. Remember, these aren't perfectly to scale, but hopefully they can help us understand. So if we had our side ratios here?
16:54
This is 45 here. This is 45 here, right? The actual angle this corresponds to would be.
17:02
Let's see. 45 degrees short of 360, which would be worth 315. I think this is our reference angle. This is the triangle we form. We have ratios of sides of. Actually, this is a positive one.
17:21
Because we're facing this direction. Except our y value is negative now.
17:27
And our hypotenuse is still the square root of two. Those are our main special right triangles. A 30, 60, 90 triangle and a 45 4590 triangle.
17:37
Now, the last places that exist on a.
17:41
Unit circle.
17:44
Are the axes themselves, the x axis and the y axis sometimes can be a little bit confusing because it doesn't quite seem like a triangle is actually happening there.
17:54
So. To imagine a zero degree angle, let's start with.
18:01
Let's say we started with a 30 degree angle, we'd have a triangle that looks like that. Right. But what happens if we were to split that England half?
18:13
And go to 15 degrees, our.
18:18
Sides would look like this. Our.
18:23
Vertical side would be shortened. And our horizontal side would be about the same.
18:30
Well, let's say we split it again and make it even narrower. And now we're getting down here. Well, now a vertical side is getting really short.
18:40
And as a matter of fact, as we keep going down and down and down and this angle is getting closer and closer to zero when we actually get to zero.
18:50
We have a triangle that's collapsed on itself.
18:53
Because from the x axis, we've gone no degrees up.
18:59
So we're actually right on the x axis. Let me make that a little bit thicker so we can see. So our triangle is going like this.
19:09
It's going this way, it's going zero amount up or down.
19:14
And then it's turning back around. OK. So in that case, the ratio of our sides is we have an X value, a side this way, horizontal value of equal to one.
19:30
Our y value is equal to zero. And our hypotenuse value, or Z or C, whatever we want to call it, is also equal to one. OK.
19:42
If we think about 180 degrees, we have.
19:46
A triangle that also has collapsed on itself.
19:51
The only difference is. Our X value.
19:56
Is negative one r y value is still zero, and our Z our hypotenuse is still positive one because the hypotenuse is always positive. OK.
20:07
The same idea holds with vertical.
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Except instead of having zero vertical.
20:14
We have a vertical that's equal to one. So now our y value is equal to one and our hypotenuse is equal to one. But our value that goes this way is equal to zero.
20:26
And if we go down. This way. We have a same idea, except our y value is equal to.
20:34
Negative one and our hypotenuse is equal to one, but our value is still equal to zero. So those are all the points that are covered by the unit circle. We're using one tool which is a special right triangle, or we're using two tools. We're using the 30, 6090 triangle and we're using the 45, 45, 90 triangle. But whatever angle you're given to work with, you can build your triangle from that because as we see when we get to actual trig functions, we want to build triangles every time.
21:04
Drawing a picture of a right triangle is going to be the most helpful tool you can do. I hope this lesson was helpful for you. Let me know if we need to cover something in more depth. Drop a comment below. Subscribe! Look forward to Lesson three. See you next time. Keep learning.
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