 Let's do a trick for a second. Let's finish off the trip. How's our timing? Oh, not bad Not bad. Let's do a little bit of trick. So we deal with that Okay Here take a look at trick Here's three pins Let me do this and bring out the Thing so I got the thing set up properly. So let's assume we have this right? Here's a triangle Okay Triangle means a polygon with three different angles, right? Close polygon So just imagine if we had this triangle, right? Now I'm going to Try not to show the logos and stuff that they got the logo all over this thing, right? I wish I had pens here or drumsticks here that I've done with before, right? So take a look at this. If I decrease this angle here, right? Which side is getting smaller? It's this side, right? If I decrease this angle here This side is getting smaller. I will search if I have an example that expresses what I mean. Okay, awesome That'd be great So the way it works is this with a triangle an angle controls the opposite side of a triangle So this angle if I draw this triangle check this out if I draw this triangle this angle controls that side This angle controls that side and that angle controls that side Okay, straight-up triangle right So if you know this just from this principle I could give you a triangle and ask you if the triangle is a legitimate triangle So for example if I give you this triangle I Say this is 50 degrees this is 40 degrees and this is 90 degrees Okay, and one of the other things here's a properties of triangle some of Angles in Triangle is equal to 180 degrees Okay, the sum of the angles in a triangle when you're quitting in geometry, which is Flat surface geometry has to equal 180 degrees. So if I draw any triangle, here's a whole bunch of triangles All right This plus this plus this is 180 this plus this plus this is 180 that plus that plus plus is 180 that plus that is 180 180 180 180 180 180 Some of the angles in a triangle have to equal 180 degrees period done Right now take a look at this So I drew a triangle here for you and the sum of the angles 180 degrees right and then I Gave you sides for the angles. I said this is six. This is five. This is nine Okay The principle we have is this an angle controls the opposite side right so this angle controls that side that angle controls That side that angle controls this side the whole side, right? I could ask you a question. Is this triangle a Legitimate triangle could it be a legitimate triangle and the answer obviously I give you one that isn't so I could prove a point, right? This triangle is not a legitimate triangle and here's the reason why Right sign laws sign law. Here's the reason why right? If an angle controls the opposite side, so this angle This angle controls this side, right? So over here this angle controls this side, which is the five This angle controls this side, which is the six and this angle controls this side, which is the nine Now the biggest angle is 90 degrees and that's across from the largest side. That's legit and then we got the smallest angle is across from The side that's six and the mid-sized angle the one that's between 90 and 40 is 50 degrees Controlling five. This cannot be a legitimate triangle Because if this is the smallest angle and if it controls the opposite side, this has to be The smallest side, right? So if I made this one a Five and this one is six and I asked you if this triangle was a legitimate triangle You would say could be a legitimate triangle. The other one was absolutely not a legitimate triangle Okay, illegitimate triangle out of here, right? Is it legitimate? This one could be legitimate now, right? So keeping that in mind Which is an angle controls the opposite side Right? I'm gonna erase this guy as well If I give you a right angle triangle all of a sudden we have three other equations that pop up here, right? another equation we have is For a right angle triangle actually we have four equations, but we'll do them one at a time, right? So this is our first property of a triangle. This is any triangle The sum of the angles in a triangle have to equal 180 degrees. If I call this beta You know, let's not call it beta because people might confuse with that Let's call it alpha. Let's call it. No, I don't want to call it alpha. I guess we'll call it alpha Should we call it alpha? Let's call this w and let's call that z Just simplify things so we don't have any problems, right? So if I give you this triangle, it's a right angle triangle now for right angle triangles You have the following four equations you can use a squared plus b squared is equal to c squared Okay, and that's straight up Pythagorean theorem which says for any right angle triangle The two legs of the triangle squared out it up equal to the third leg squared, right? It's pretty common to pair a and alpha b with beta. Yeah, I put that one theta No See with gamma see what goes with gamma racer kill by the way, I raise a kill I didn't realize see what with gamma, but we're just gonna call w and z doesn't make a difference, right? The beta should have put here. That should be alpha and then beta and then gamma was a symbol for gamma. Yeah, that's this I forget Pythagorean theorem a squared plus b squared equals c squared So if I give you any Right angle triangle as long as you have or angle c opposite of c. Yeah, you know what? Let's call it that here We'll do this We'll keep it and I really dislike the convention they use by the way. I don't Personally it's not my favorite, but basically what they do they say the capital letters are the angles and The small case letters are the sides Right Yeah, that's what they do. I'll make the C a little bit smaller. So it shows as a small C right now when it comes to this situation and by the way for any triangle if you have three of The pieces of information because there's six unknowns in a triangle There's three angles and three sides, right? If you have three of the bits of information One of them has to be a side. You can find everything else. I Think small and capital is good, but it's annoying For letters like a very annoying for letters like C very annoying because C looks just like C except for that except for size I don't like it personally, but it's really it's the convention that For some reason it's been being used for the last 15 years or so, which is weird, right? so if I give you any right angle triangle or any Triangle really and if I give you That one two five x w z And if I ask you to find this you can do it if I give you this triangle Right angle triangle and I say five seven two I didn't give you any angles except for the 90. Oh, I don't need to even give you that one. Oh, that's the same as that one I Actually, this doesn't even have to be right angle triangles. You can figure that one out, right? But if I give you this here, let's call this 50 degrees and that's 90 You could figure out that stuff if I give you kill that if I give you this you can figure out the rest but if I give you this 40 if the sum of the angles in a triangle is 180 then 180 minus 90 because that's 90 is 90 minus 40 it makes us 50 if I give you this three You can't figure out this triangle because you don't have The scale right you don't have a distance. You don't have a length Okay, so as long as there's six bits of information in a triangle three sides and three angles As long as you have three of them one of them being a side you can figure out all the rest, right? So For right angle triangle Pythagorean theorem says this The legs of the triangle and the legs are the ones that connect up to give you the 90 degrees The legs of the triangle each one squared out of together is the hypotenuse squared The other three angles give trials of the scale. So you know the ratio. Yeah, you know the ratio That's it Racical thank you for the clarification. So check this out if I give you A triangle if I say this is 30 degrees if that's 40 degrees 30 and 40 is 70 degrees Right subtract that from 180. So this would have to be 110 degrees, right? If we have this triangle The ratios of the sides We know Right because no matter how small or how big this is the ratios side a b and c A over b will always be the same Here's another one And if I said this is the same angles 30 40 and 110 degrees and this is x y z then a over b has to equal to x over y z over y Has to equal c over b Right so proportions matter as well and they come in very very handy Okay Triangles are very they got some pretty cool features to them They got pretty cool features to them Here's the trigonometry aspect of the formulas, right? Sine theta which is basically they give you the sine theta is opposite over hypotenuse Coast theta is equal to adjacent over hypotenuse and tan theta is equal to opposite over adjacent Now what does this mean? This means the following The theta is the angle where you're at And relative to the angle that's what the opposite in the hypotenuse And the aj center going to be the hypotenuse is always across from 90 degrees, right? so if The theta is here Then sine of b sine of b Is equal to opposite over hypotenuse so opposite from b is b over the hypotenuse is c Coast of b Is equal to You put yourself whenever they're giving you an angle just imagine yourself Putting yourself there And cosine is adjacent over hypotenuse, right? So that would be a over c because that's the adjacent Over c now if you put yourself here This c the hypotenuse is also adjacent to b Which is a little confusing when you they call it adjacent over hypotenuse c is also adjacent to b but c already has a name it's called a hypotenuse So you don't refer to that as the adjacent or the opposite. That's always the hypotenuse I still don't like this. They're being 360 total degrees it feels like a product of the Imperial system. It's You mean for a full circle 360 degrees It's based on the sun It's based on I looked into this a while ago So I want to ask this question before and we looked into it and there was a reason for it Which was a pretty legit reason I forget what it was. Is it based on the sun? So coast of b Is going to be adjacent over hypotenuse and over c tan of b tan of b Is going to be opposite over adjacent. So you put yourself at b and you go opposite b over a b over a On the same note One reason is that this is highly divisible Yeah, would you rather it be 2 pi? It is radians 2 pi radians right and 2 pi comes in way more handy than 360 degrees right Like radians is way better than degrees but degrees because when taught forever We just relate to degrees better Right for in grade 12 they change it up to or grade 11 if you're lucky you get a teacher They introduce pi and all of a sudden you're like what? Yeah, the old mathematicians looked at the sun and figured that the sun took approximately 360 days around us Is that what it was? That's simple as that Damn That's a stupid reason to make a circle 360 degrees But I guess at the time at work. You didn't have calculator. You got to keep in mind You didn't have calculators way back then, right? So you had to simplify things as best as you could The more you know the more you know the more you realize how ridiculous our world is, right? Let's say we wanted to find sine, cos and tan of angle a So if you go sine of a Sine of a sine of an angle is opposite over hypotenuse So what you do is you put yourself on a oops You put yourself at a and you go what's opposite from a From here the opposite is here and the hypotenuse is here. So sine of a is a over c cos Of a oops cos of a is b over c And tan Of a is a over b Now take a look at this a over c Where else do we have a over c? a over c So what that tells us is cos of b is equal to sine of a Right, these are sort of properties that pop up Levia, how are you doing? Welcome. Welcome to the last stream mathematics, right? b over c Oh, b over c So sine of b is also equal to cos of a Right, this is some stuff that we delved into during our trigonometry Playlist, right What we're dealing with more how they're related to circles and what not, right? So this is what Sokotoa represents And the best way to learn is to do a problem. So let's do a problem That way you'll see or a couple of problems. You'll see how this plays out with real numbers, right? Oh, I erased the tan. No, I didn't erase the tan. Cool So let's assume we had the following Triangle and we wanted to find We want to solve the triangle whenever they say solve a triangle. They mean find all six Properties of that triangle. Okay So here's question one and we're only going to deal with right angle triangles. Okay So I've given you a right angle triangle. Let's assume I make this a 10 and I make this 60 degrees. Okay, I want you to find x y and z Okay, perfect timing I'm covering exactly the same topic now. Awesome. Olivia. I hope it helps. Here's an example, right? So what they would say for this, they would say solve the triangle solve this triangle Okay Apologies about my writing. I write like a Screw me, right So my question to you is and whenever you're doing these types of problems You should ask yourself. What do you want to solve for first? Right now? We have Choices we could solve for any of these first, right? Sometimes you don't have a choice. Sometimes you have to solve for one Before you can move on to the other two. Okay But right now ask yourself, which one do we want to solve for first? Okay, have you figured it out? Do you know about south calculator? I've actually made it 60 degrees because that's a special triangle We know the ratios. We can actually do this without a calculator. Okay It's easy zed Zed very good, right? A Ali cat because that's the easiest one to figure out Why is it the easiest one to figure out because we have five formulas, right that we use for right-angle triangles two three four five right well Four formulas that only work for right-angle triangles and one formula that works for all triangles, right? The first formula says the sum of the angles in a triangle is equal to 180 degrees Doesn't make a difference. What type of triangle you have as long as it's on flat surface triangle You can Figure it out. It follows this principle, right? So some of the angles this angle plus that angle plus that angle has to equal 180 degrees Now a lot of people do this. They always add Because this is a right angle. That's 90 degrees They always add 90 degrees to 60 degrees and then subtract from 180 If you know, this is 90 degrees 180 degrees minus 90 is 90 So whenever you have a right-angle triangle If it's a right-angle triangle the sum of the other two angles has to equal 90 degrees So you don't have to add this to this and then subtract from 180 You just have to go oh if this is a 90-degree triangle then this plus this has to equal 90 So 90 minus 60 z is 30 right So we figured out z This is 30 degrees That was easy, okay Now we have a choice We had a choice before but because we only got two sides left Our choice is do we want to solve for x first or y first? Which one do we want to solve first? x or y x or y What should we solve first? Difficult choice. Should we choose one? Let's start out with the lowest alphabet right x. Let's solve for x first So if you're going to solve for x first Opposite opposite, which one? Here's the key, right If you're going to solve for x You have all three angles, right? So z was 30. Let me put this here. So we know It's this angle here. It's not sitting out in the middle of the triangle 30 degrees, right? x is easier if you know sign of 30. Yeah, why is the same? It's going to be the same, right? But let's assume we want to solve for x That's your first question the decision or your first dilemma Which one do you want to solve for first or which one can you solve for first? And then when you figure out it's x you want to solve for you're going to ask yourself Do I want to use 60 degrees or do I want to use 30 degrees because you can use both of them right now? Okay, if you're going to use 30 degrees you ask yourself, what's x? What's the position of x? It will be half it won't be half What's the position of x relative to 30? Well x is opposite from 30 So once you decide you want to solve for this and then you decide you want to use 30 degrees Take your pen and go I'm at 30 degrees and I'm going to solve for x So what's the position of x relative to 30? That's opposite And what length do I have right now? Well, I have the hypotenuse So I'm looking for something that has opposite and hypotenuse if I'm going to use 30 Opposite of hypotenuse is sine So this would be sine of 30 degrees Is equal to opposite opposite x over 10 Okay, and if you cross multiply Like looking at the horizon like looking at the horizon You cross multiply this up. So x is equal to 10 Sine of 30 and sine of 30 is a special triangle 30 16 90 1 square root of 3 2 sine of 30 is 1 over 2 So this would be 10 times 1 over 2 You can punch in your calculator if you want which is 5 So you just figure this out. That's 5 Okay Now keep in mind You could use 60 degrees to find x Here I'll keep that one up the special triangle Yeah, special triangle is just a ratio. It's a set Ratio it works, right We talk a lot about this in the trick playlist If this is 30 degrees that's 60 degrees. That's 90 the ratios of the size are these So let's assume we're not going to use the 30 degrees We're going to try to find let's assume we haven't found this yet We want to find x but we're going to use 60 degrees to find x and personally I usually always Use the information they've given me instead of the information I've calculated to do the next calculation Just in case I made a mistake calculating this then the mistake doesn't carry over to the next problem right To the next unknown you need to solve for So if I'm going to solve for x I'm going to use the 60 degree angle So I'm going to put myself at 60 degrees right And I'm going to say okay, what side do I have? Well, we have the hypotenuse So I'm looking for it's going to be one of these guys because they both have hypotenuse and I'm it's not going to be 10 Because it doesn't have hypotenuse So I have the hypotenuse and what's x relative to 60? It's the adjacent relative to 60. So I have to use cosine so this becomes cos of 60 degrees is equal to adjacent adjacent x over 10 cross multiply x is 10 cosine of 60 Okay, what's cosine of 60 you put it here cosine of 60 is adjacent over hypotenuse 1 over 2 So this becomes 10 times 1 over 2 which is equal to 5 This is equal to 5 right and you notice cosine of 60 is the same thing as sine of 30 All right, these are properties that show up cool I hope that's okay. I talked like mad in this stream craziness craziness fun stuff though and trigonometry is ridiculous and important It's really ridiculously important once you know how to solve use so katoa the Pythagorean theorem to some of the angles are 180 degrees Two months of grade eight nine you'll ace Okay, and you're set up for math 11 and math 12 and you begin to have a Way better appreciation for what all of this stuff means and it's ratios Okay, let me get us thumbs up It's ratios of one side relative to another side. Let me explain to you what that means Okay, I'm going to erase these and if you do need the stuff you can Definitely on the video. We're going to load it on youtube and it'll be available on twitch I think for a couple of weeks. You can definitely take screenshots of it if you learn basics of this It's so easy to solve all problems. Yeah, basically a lot of problems All right, so take a look at this. I'm going to erase these guys now