All right, class. So we're going to continue on from last lecture where we talked about kind of the basic concepts of, like, you know, what is angular kinematics, what angles we're going to use for this. And again, the biggest thing that I need to stress is, like, being able to calculate and understand angles.
So we're going to show a couple example questions here on just how we can actually calculate these angular kinematics using, like... coordinate plane, which a lot of the example questions are going to be given are going to be on a coordinate plane like the one we see here. So we have a coordinate for the hip, knee, and ankle.
And the question is saying, given the fallen coordinates, calculate the knee joint relative angle. Remember, relative is the joint angle. thigh segmental angle or the absolute angle. Okay.
So that's just relative to the horizontal. So knee joint angle is going to be just the angle of the joint itself between the hip or the thigh and the shank or the tibia, whatever you want to call it. And then the relative angles or the absolute angle is going to be relative to the right horizontal.
Okay. So let's start with the knee joint angle. We're going to label that as angle B here. So there are two different ways we can solve for this.
We kind of showed you one already last week using law of cosines, which I'll show you again in a second. But I want to show you a different way that makes a little bit more sense in my head to do. So when we have these big, like, you know, not right triangles, obviously this is not a right triangle. You can't create a right angle on here. It's over 90 degrees.
We can actually use... or break this into two separate right triangles in order to solve for our answer here by creating this imaginary right horizontal right through the center of the big knee angle we're looking at the b knee joint okay so this breaks it into two different triangles as you can see here we have the top one which is using a theta angle and then the bottom one is using an alpha angle And if we add this theta to this alpha, we can calculate for beta. OK, but first things first, we need to, you know, figure out what is this angle and what is this alpha angle? How do we do that? Well, we have two separate right triangles here.
We can start with the top portion. This is the thigh. So what we can do is we can create this imaginary point on this coordinate that we can use to create. a right triangle does so.
As you can see here, now we have a right triangle. We got this coordinate by using the five. So you see here, if we look at this, this is the x coordinate, this is the y coordinate.
So if we look, just go straight down, the x coordinate's not changing. So we're going to keep it as five. The y coordinate is going to be used, or we're going to use this six right here. Okay, again, because on this, when we're comparing it to this coordinate, we're not changing the y. It's not changing in height.
It's in line with this coordinate plane. So we're kind of mix matching or kind of combining these two points to create a new point that's in the center of both of them that creates this right triangle. Now what do we do next?
Well, if we have this, we're looking for this angle right here. We need to. Yeah, I should have just looked ahead. So we're looking for this angle right here.
Well, we can use either distance formula where we use distance formula to calculate this side, this hypotenuse. Or we can just know that this drops down four units. So you can see here the fives are the same.
Ten and six are different. So let's say it drops down four units. We look from the left to the right on this coordinate plane looking for this. I guess adjacent side, we're going to say it's going to travel six units. And we'll get into that a little bit more and then how we use tangent in order to do this.
But let's look at the bottom side and how we can create a right triangle out of this. So we have two coordinate points. So this is the shank. We have the 11 comma 6. We have the 9 comma 1. Now we can create this new point in the middle of these two. It creates a right triangle as shown.
This is going to be 9, 6. This is going to be 9 because, again, it's in line. The x-axis is in line. with the bottom portion. So we're going to use this X value up here.
And it's in line with the horizontal portion of this value right here. So again, we're going to take the nine from here, throw it up, six from here, throw it in. And now we have a new right triangle where we're solving for this angle right here.
You remember, this is the same exact thing. And we're solving for that angle that's in line with the kind of combines to make the knee angle itself. Okay, so let's look at the first portion and how we can use trigonomic functions in order to figure this out. So again, we can use distance formula to find each side angle, but I already showed you that because the vertical side equals the square root of 5 minus 5, square plus 10 minus 6 is going to equal 4. So again, if we use distance formula, it's going to give us 4 anyway, but we can just say we know that the 5s are the same, so we can just say the two things that are different. 10 minus 6 is going to be 4. So if things are in line with each other on the horizontal or the vertical axis, you can just subtract and add.
We don't have to use the distance formula if you don't want to. You can. It's going to give you the same value. It's just going to be extra work.
I don't like extra work. It's called a parsonomic. Basically, trying to find the easiest solution to a question. So again, we can do the same thing for the bottom one. So now we have this angle 4, this angle 6. We can label this as the opposite.
This is the adjacent. We can calculate for the hypotenuse if you want to, but we don't need to. We have two sides.
You have two sides of a triangle. You can always solve for the angle you're looking for using tangent in this case because tangent is opposite over adjacent. So opposite over adjacent is going to equal 4 over 6. So tangent beta, which is what we're looking for, equal to 4 over 6. And then we use the opposite function.
So this 0.66 is going to be 4 divided by 6. So we can just use opposite function. 10, opposite 1, 4, 6 equals theta. And once you do that, it's going to give you the answer.
Typically, if your calculators are set for degrees, which they most likely are, It's going to give you an answer in degrees. Okay. Now we do the same exact thing for this. So we know, whoa, let me zoom in.
So we have this two on top. Again, that's the difference between 11 and nine. It's going to be two. We have six and one.
That's going to be five. So that gives us this side and this side. And again, now we can use tangent again.
But the opposite side is going to be this five. The adjacent side is going to be this. 2. And we just solve using the exact same formula, opposite function tangent, in order to get a new answer of 68.2 degrees.
Okay. Now we have both sides of the triangle. We have this 33.69, which is the angle right here.
And this alpha angle, which is 68.2, which is the bottom one. And both of these angles combined together is going to give us the total knee angle, this beta. right here.
So all we have to do is just add them together, and we get an answer of 101.89. Now let's look at a different way to do this. Let's look at the law of cosines, which is something we talked about last week. We've given you this equation already. It's on your equation sheet, so you should know how to do this.
So again, I'm not going to go through this. This giant explanation right here just leads to this. And if you look right here, It's kind of a little different than what we were showing before. So a squared equals b squared plus c squared minus 2b times c cosine alpha. Interesting.
So just know that whatever is on this side and this side are going to have to be opposite to each other. So if we're looking for this angle right here, this alpha has to go in this spot in the equation. The c and b can be mixed and matched.
Doesn't matter. It's just whatever is opposite of the angle we're looking for has to go on this side of the equation. It has to be plugged in for this value on the other side. And as you can see here, there's only one of them. We only have one A in this equation.
We have two Bs, two Cs. So the one that is the only, like there's only one variable for it in the equation has to be the one that's on the other side of. the angle that we're looking at okay So if we were looking for this angle right here, C would be going in that spot.
We're looking for this angle right here. It'd be B that was going in that spot. OK, so just depending on the angle you're looking at is when you're going to be using this. And you can use this equation for any triangle that's not a right triangle.
OK, if you're trying to look for the angles. So let's kind of figure out how we can do that. So it gets kind of reworking what I was just talking about. We have a C equals this side right here.
B equals this side right here. Depending on what we're looking at, this will vary, okay? So, we have this coordinate plane. Let's just say this is the knee.
Again, we're using the same exact equation we did last, or the same coordinates we did last time. So, we have an A, B, and C. You can label this however you want. Just know, we're looking for this angle.
This value over here has to be plugged in right here, okay? I know people are going to get confused on that, but try and just remember if it's the angle on the opposite side of what we're looking for, this angle right here has to go in the first variable. Okay, so let's plug this in.
So we have 5, 10, 11, 6, 9, 1. So we need to get all the sides. the distances between all these segments. We have this segment A, segment B, segment C.
So B is the thigh, C is the shank, A is just this imaginary line that we created in between these to make it a triangle, okay? So you can calculate all of these. A is going to be 9.85, B is going to be 7.21, and C is going to be 5.39, okay? We'll plug these in, 9.85.
7.821, 5.39. And the way we did this is just using distance formula, okay? Get familiar with using distance formula. And now that we have these, we can just plug and chug. Plug them all in, and we just use some algebra here in order to get an angle of, look at that, 101.88.
So the exact same as when we did the other direction where we... you know, split it up in the middle and solve for the top triangle and the bottom triangle. Either one of these will work.
It's whatever you want to do. This kind of benefit of this is you don't have to kind of try to geometrically figure out two different triangles. You just have one. You got to solve for the sides.
The other one is nicer because you don't have to do a bunch of distance formulas, which I prefer. I can do the other equation a lot quicker in my, you know, I can work it out in my head as well, a little bit easier. So again, don't have to, either one will work.
It's your decision on which one you want to use during the exam. Okay, so given the following coordinates from the hip to knee, this is the original question we asked. We just solved for that knee joint angle. Now what it's asking for is the thigh segment angle, which is going to be this side right here.
Okay, so that's the thigh. What is the thigh segment angle or the absolute angle? It is going to be this angle. Whoa, why isn't it doing it?
Boom. Right here compared to the right horizontal, okay? And you can place this kind of anywhere you want.
It's going to be the same angle, okay? Okay, you don't have to place it all the way down at the bottom. You can place it kind of throughout this.
Whatever works better in your head when solving this, okay? But we're looking for this angle right here that's relative to the right horizontal. How can we do that?
Well, we kind of did something similar before, and we were looking at that, you know, kind of continuation angle. You guys remember that? We can create a new angle right here, and we already did this when we were solving for the knee. If you remember, that's, this is the knee right here.
We solved for this top portion, which was this alpha angle. We called it alpha angle. So we already know that 33.69 based on what we did already.
But again, if you don't know this, if you didn't solve this already, you can solve it again by just, you know, creating this imaginary kind of like continuation angle onto in order to figure out kind of what this is right here. And if we know the whole thing is 180 degrees because we know a flat line is 180 degrees, what we did is just kind of combine this, made this into a flat line. Solve for this angle. So the remaining portion is going to be this. So if 180 equals the total, of a plus theta we know what a is we know 33.69 we're looking for this theta value here we just have to subtract subtract 180 minus 33.69 and we're given 40 146.31 that's going to go right here okay so again we just arbitrarily chose this marker right here in order to solve for this there's other ways you can do this you can even solve if you want to make a you know, triangle here, which is something you can do.
You make a triangle on this side of it, solve for this alpha angle or whatever angle you want to call it here, and then just add it to 90 degrees because 90 plus whatever that angle is you're finding there is going to equal the total. Sorry this looks like a total mess. I hope you guys follow along, but there's multiple ways of solving for this. And again, as I talk about in this class, there are always multiple ways to solve for these equations. Okay.
All right. So let's just do the same thing for the shank, just to, you know, practice. I'm going to give you guys a second.
Go ahead and pause if you want to. Let's try to solve for this. Okay.
So the way I would do it is I would basically create a new triangle. Okay. Sorry about that.
My video stopped for a bit there. My iPad isn't working right now, so I'm going to try to draw out a little bit of this. But I think a lot of this is what I'm going to have you do is just kind of do this on your own. So given the following coordinates for the hip and the knee, the shape segment angle. So again, what we're doing is the segment angle, which is the absolute, meaning it's relative to this horizontal.
So we're looking for this angle right here. To do this, you create this imaginary dot right here. You don't have to do it on the other side of it. You can. Kind of like how we did the last triangle.
Like, you know, we're looking at this angle. We made a triangle over here. You can do that, but it's a little bit simpler if we just look at it on this side.
I'm not going to do the math here, because it would be really tough to do on this pen. But. practice these problems as much as you can.
We have one more lecture that we're going to go a little bit more into some other practice problems. So.