 So what we're going to do is convince ourselves that the ratios of the sides of a triangle don't change as long as The value of the ratio anyway as long as the The angles stay the same as long as they're similar triangles Okay, so let's take this down and just convince ourselves really quickly what it is What I need to do is take pictures before we take this down So let's take Pictures of this thing That way we can use it too bad. I mucked up this 45. Otherwise, it would look really pretty well pretty ish Considering my handwriting and this special So let's take this guy down Great papers that we took down and put it on the side So it can be as accurate as possible So let's do a little calculation. Let's take a look at the similar triangles and Convince ourselves that the ratios the trigger the trigger ratio cycle sign attention Don't change no matter what the sides how big or small the triangle is as long as the angles stay the same And this is something we did in Series one we took a look at similar triangles, right? So what we're gonna do we're gonna generate a triangle Or a couple of triangles anyway, so let's take this and So I'm gonna go from here see how far we're going One two three four five six seven eight nine nineteen. Let's take it to twenty one two three four five six seven eight Have a triangle here. Now what I'm gonna do. I'm gonna measure these distances. This is one two three four five six seven eight nine Ten so this is ten here Right. I actually don't care what these angles are so that's the distance of ten and For us to prove that the ratios don't change no matter how big or small we go with this thing We don't care what the angles are because if it works for one Triangle with the ratios right it's gonna work for the other ones. So if this is ten Let's take what was this one fifth twenty. I believe one two three four five six seven eight nine ten Sixteen seventeen eighteen nineteen twenty so the distance here is twenty Right this angle is whatever angle it is Angle a this angle is whatever angle it is and that's 90 degrees Now what I'm going to do I'm gonna create a couple of more triangles Actually, let's just do yeah, let's create a couple of more one couple other ones So I'm going to go to here And draw a line here because it's out across errors this distance here is one two three four five six seven And this distance here is one two three four five six seven eight nine ten eleven twelve fourteen So this distance here is fourteen The next level up And this guy goes here To here I could create another one here if I want Okay, and this distance is from here if I create another triangle Let's make it here. I guess One two three four sure So this is four and this distance is one two three four five six seven eight So this guy becomes eight Now what I'm going to do is we're gonna Take a look at this convince ourselves just using ten of This angle because that way we've got exact Grids on here. We're not going to go diagonally because that complicates things. I don't have a ruler to measure it So if it works for 10 theta, it's going to work for cos theta. It's going to work for sine theta, right? It has to be consistent so 10 of this angle let's call this angle a Okay, and this is angle b So 10 of this angle for the big triangle is 10 over 20 10 of a is 10 divided by 20 okay Well 10 of this angle also happens to be seven divided by 14 so Is seven divided by 14 10 of this angle also happens to be four divided by eight four divided by eight Now 10 divided by 20 is a half Seven divided by 14 is one over two same deal Four divided by eight is one over two same deal, right? One over two So 10 of this angle Is one over two the ratio Of the opposite side relative to the adjacent side is one over two For this triangle This triangle and this triangle because All three of these triangles Are similar triangles. They've kept the same angles one. It just happens to be A bigger version of the other and a bigger version of the other, right? So when we're studying our special triangles Right, we had our distance for a special triangle for 45 45 Right, we had one one root one one the square root of two Right, that doesn't fit on a unit circle because the radius for unit circle is one But we don't care because we're not looking at The exact values Right for the size when we're Doing the trig ratios when we're looking at sine cosine and tangent because sine cosine and tangent are ratios of one side versus another side, right? So Irrelevant of how small the circle is or how big the circle is right The ratios the sine theta cos theta and tan theta of the size of the triangles of the right angle triangles Or the coordinate systems They don't change the ratios are the same which is should be obvious by this right if this The seven unit one was our Circle right Unit circle then if we go bigger the ratio doesn't change We could figure out what the distance here is if we do Pythagorean theorem should we do? I don't know a lot of work, but let's do let's take a picture of this Before we do the Pythagorean theorem so if we want to Convince herself that this also works for sine cosine and tangent What we could do is just use the Pythagorean theorem to figure out what the distances are From there to there from there to there and from there to there and You can do that as an exercise if you like and if you want we'll do it for what we have to do is do it for Two of them to convince ourselves, right? So let's do it for the big triangle and the little triangle So if we do it for the big triangle Pythagorean theorem is a squared plus b squared equals c squared for the big triangle is going to be 10 squared plus 20 squared is equal to c squared So 100 plus 400 is equal to c squared. So c is going to be the square root of 500 that's the length of The big triangle right of this guy squared of 500 If we're going to do it for the little guy, it's going to be 4 squared plus 8 squared And we're going to take the square root of it, right? So this guy is going to be 4 squared is 16 8 squared is 64 so 16 plus 64 64 plus 16 0 1 7 80 so square root of 80 Is this distance here? And if we're going to look at let's say sine of a For the big triangle is going to be 10 divided by the square root of 500 So sine of a Is going to be 10 divided by the square root of 500 And sine of a also happens to be 4 divided by the square root of 80 4 divided by the square root of 80 Let's bring out our calculator and this is calculation. Make sure this is The same value right which it should be it's a solar calculator old school. So I need a little light to get this thing back on So let's go 10 oops 10 divided by 500 Where's my square square root? Is equal to 0.447 right so 10 divided by the square root of 500 is 0.447 That better be what 4 divided by the square root of 80 is right? So 4 divided by 80 second function square root Is equal to 4 0.447 dot dot dot So same number the ratio didn't change right So if one of these circles one of these Triangles Happened to be our unit circle then The other Triangle would have been a bigger circle representing a bigger circle It could have been Our special triangle 45 45 90 or 30 60 90 So the ratios the sine and cosine Didn't change. Okay. I hope that's clear. It's something that Some students sometimes ask me and I do go through and explaining them taking them back to great 8 in great 9 and as soon as I draw this They're like wow That's the reason well one of the reasons we studied this and My answer is yes, but it took You know if you studied in grade 8 it took four years three or four years until You appreciated how powerful this thing is when you start talking about the trigonometry trig ratio sine cosine attention. Okay, so That's the two special triangles that we have to know. Okay 45 45 90 and The side links are one one root two Okay, from there you can generate your trig ratios And the other special triangles 30 60 90 And the side links are one Squared of three and two and from that you can generate your trig ratios And we're going to dive a lot deeper into this in the next video because what we're going to do We're going to take our unit circle again And we're going to look at all our stops For these special triangles. So what we're going to do Is move around the circle and our first stop is going to be 30 degrees next stop is going to be 45 Next stop is going to be 60 90 120 and so on and so forth And we're going to generate the table for what the trig ratios are For those angles I'll see you guys in the next video. Bye for now