welcome back to class in today's lesson we're going to be talking about Pythagoras's Theorem and trigonometrical ratios our objectives are to describe Pythagoras's Theorem to describe the primary trigonometrical ratios to find one ratio given another ratio and to describe relationships between the ratios first up is Pythagoras's Theorem now what exactly is it it is an idea that has been around for a really long time as far back as the ancient Egyptians who built the pyramids essentially what we do here is that we're taking three squares and playing around with them to use them to form a right angled triangle in doing that you realize that once you get a right angle triangle to form that this the square on this side plus the square on that side is equal to the size of the square on this side that is the length of this side of the triangle squ the length of this side squ isal to the square of this side essentially that gives us Pythagoras's Theorem Pythagoras's Theorem Works only with right angled triangles and is used to calculate unknown lengths in this triangle here we have a situation where this side is three three units whatever the length of the unit is this side is four units and this side is five units so if we should write it down this side 3 SAR + 4 squar would be equal to the longest side which is 5 square there's a reason this side is is the longest side it is the longest side because in this triangle this angle here is the largest angle the right angle is the largest angle and so the side across from that angle or opposite to that angle is actually the longest side in any triangle whichever side is longest is usually opposite to the side the angle with the biggest value so here we are this side is three this side is four and this side is five so 3 S which is 9+ 16 which is 4 square is equal to 25 and as you add these 9 + 16 you'll realize the 25 is 25 so using letters a b and c we can say then that if this side is B that side is a then a s + b² is equal to C squ or whatever letter in convention you want to use but this is essentially the idea of Pythagoras's Theorem we use it to calculate unknown lengths as I said earlier there are some things that are called pythag and triples and this triangle here we call it the 345 triangle because one side is three one side is four one side is five is a very very special one and it tends to come up on a lot of questions in your C exam often not the multiple choice if not this one then a multiple of this one they call them Pythagorean triples so for example if we should multiply this Triple Three numbers by a number two for example then we would get 2 * 3 is 6 2 4 is 8 2 5 is 10 and so on and if you should multiply it by three then 3 3 is 9 3 4 is 12 and 3 5 is 15 all of these numbers work with the setup that we have here so 9 s + 12 square would give you 15 squ and 6 sare + 8 square would give you 10 square it's just a nice thing to know sometimes if you know these things going into an exam and you see the question then you don't have to stress about working it out so how exactly does Pythagoras's Theorem work in terms of calculating sides um we can calculate pretty much any side using the two sides that form the L and here's our L those are what we tend to call the shorter sides um the sides that form the L means that this the sare square of the sum of the square of those two sides so in this case we're going to do it 3 S Plus 4 square is equal to 5 Square remembering that the idea behind Pythagoras's Theorem says these two sides square a square + b s is c² and so in our case here we're going to have X because X here is the hypotenuse this is the 345 triangle but we're going to work it out just the same to show so 9 + 16 gives us 25 and adding those 25 is equal to x² and now we need to find our square root so the square root of x² gives us the square root of 25 so X here gives us five in this example example here using the 68 6 81 so we already knew that this was 8 but just to show then these two sides make the L so you could write it down as x² + 6 s gives you 10 s which means x² + 36 is equal to 100 you do the subtraction 100 - 36 would give us 64 and and so X would be equal to the square < TK of 64 which is 8 and pretty much this is how we use Pythagoras's Theorem to find the length of one side given to other sides when we mark an angle apart from the right angle so in this is our right angle triangle this is our right angle it is marked so we can know that which which angle is a right angle and which side is is the longest side when we when we mark an angle that is not the right angle as in this case we marked this angle we call it Theta it's a Greek letter then we can give the other sides of the triangle names the longest side of the triangle has a special name we call it the hypotenuse sometimes we just refer to it as the H for hypotenuse the side that is in front of the angle here's the angle so the side that is in front of the angle or directly opposite to it we call it the opposite sometimes o and the side that lies next to the angle is the adjacent side sometimes we call it a now um the side BC is also lying next to it these two sides are lined next to the angle but this one already got a name we call it the hypotenuse because it is the longest side special name the other two sides are given names based on where this angle is so because this angle is here Theta this side which is right across from it is called the opposite side and that is what it means in mathematics when we say opposite we say directly across from so the opposite side is AC the adjacent side is AB and the hypotenuse is BC let's look at another example where the angle is in a different place now the angle is up here this is your right angle which means that here is your hypotenuse so the hypotenuse is DF this side is directly across from the angle right in front of it because of that it's called the opposite so EF is the opposite and the other side is the the adjacent side this is actually very important that you learn how to name the sides based on your position naming the sides leads to choosing the right tools for the job that you have to do so the angle position determines which side is opposite because the angle is here straight across that's the opposite and we know that once we have a right angle marked the side straight from it is called the hypotenuse and the other side is called called the adjacent side now when we compare two sides at a time in relation to an angle that is marked we get what we call trigonometrical ratios there are six ratios let me point that out early but for C cxe purposes for cc purposes especially only three of them are concentrated on so there are six but we only concentrate on three at the C6 level so when we comine two sides at a time so we compare them a ratio is essentially a comparison so when we compare two sides at a time and in com in in relation to the angle that is marked we get these three ratios so we call this one sign so we say the sign of the angle let me write it further out sign this is how we write it in short and this is how you would see it on a calculator so we say this sign of the angle this is the angle is equal to the opposite side this is the opposite side divided by the hypotenuse and this side is the hypotenuse so s is the opposite side divided by the hypotenuse we say the cosine we shorten it as C it's cosine c o s i n e the cosine of the angle the angle is always there is equal to the adjacent side so we take this side and this side together a over H and the last one we call the tangent and the tangent of the angle is equal to the O over a as in O / a so you take two sides at a time with the angle the first two take the opposite and the hypotenuse and we call that sign take the adjacent and the hypotenuse and you call that coine take the opposite and the adjacent and we call that tangent and though these are the three primary ratios actually these are more primary than this one because there's a relationship between these two that actually gives this one but generally we tend to ref to these three as the primary ones or the main ones all right now you notice out here that you have three other things so the sign of the angle is the opposite over the hypotenuse or the opposite divided by the hypotenuse the angle itself is what is to find an angle we use what we call the inverse function so this is a function it takes in a value it gives out an answer this is the inverse of that and the inverse takes in a number and gives out an angle the cosine will take in an angle gives out a number the inverse cosine function takes in a number gives out an angle same thing for tangent you put an angle at tangent tangent of an angle it gives you a number the angle here would be equal to the tan inverse and as I'm seeing it here this should actually be tan inverse and not C So Tan inverse would be opposite over he over adjacent or all right moving on now just as as just as important as it is to learn to name the sides correctly it is also important to learn how to write down the ratios correctly so here we are going to write down for each triangle the sign of the angle the cosine of the angle and the tangent of the angle before we do that we need to name the sides so the angle is here which means that this side straight across from it is the opposite the this is the is the right angle so the side right across from that is the longest side it is the hypotenuse and the other side is the adjacent because we write s of angle is equal to opposite / hypotenuse and cosine of angle is equal to adjacent / hypotenuse and the last one tangent of angle angle is equal to opposite over adjacent some persons long ago not sure who they were kind this phrase called SOA some students find it easy to use to remember the three ratios so so meaning sign is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is opposite over adjacent so in this triangle here we can write down the of the angle the co of the angle and the tent of the angle the sign of the angle the angle is Theta is equal to the opposite over the hypotenuse which tells us that the S of the angle here is equal to 3 over 5 the coine of the angle is equal to the adjacent side ID the hypotenuse and in this situation the coine of the angle the adjacent side is four and the hypotenuse is five and the tangent of the angle is equal to the opposite side divided by the adjacent side which gives us that the tan of the angle is z equal to the opposite side is three and the adjacent side is four in this triangle here's our angle here which means that this side over here is the opposite this is our right angle which means this one over here is the hypotenuse sometimes your questions will say that your triangle is not drawn to scale sometimes they'll draw them really weird and you have to use the fact that the right angle is opposite to the hypotenuse to to see them because they won't look always look at the lens that they supposed to be and because this is opposite and hypotenuse this one up here is adjacent so we can write them down as s of the angle of this angle is equal to opposite over hypotenuse which is POS is 15 over 17 the cosine of the angle is equal to the adjacent side / hypotenuse so it's 8/ 17 and the tangent of the angle is equal to the opposite side divided by the adjacent side which is 15 over 8 finally let's look at this one again the angle is here so we want to know which side is opposite the side right across from it is opposite so this one is opposite it's is important again like I said to label them correctly otherwise once you start label them wrong you are going to run into errors because naming the sides incorrectly will lead to you choosing the wrong ratio to work the question so this is the opposite here's our right angle which means that this is hypotenuse and here is the other side that we call the adjacent side so again so sign is opposite over hypotenuse so the sign of the angle you must always say the sign of the angle whatever the angle is is equal to opposite which is 2 over 2.5 the cosine of the angle is adjacent over hypotenuse which is 1.5 over 2.5 and the tangent of the angle is equal to the opposite side divided by the adjacent side which is 2 over 2.5 the place where you're most likely going to be asked to write down a ratio it is most likely on the paper one on the multiple choice paper where they give you a question and ask you to State the sign or cosine of an angle um on paper two you'll most likely be using it rather than just stating what it is there are some special relationships between the racials and we're going to explore explore them a little bit here and some of them that you should know it's not required that you must know them but it's nice to know them and the the relationships are quite unique some of them even cute um sign of 30 and when we say 30 we mean um 30° and 60° so let's write these down the sign of 30 to remember so s is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is opposite over adjacent so the sign of 30 here's our 30 this is our 30 it means that in relation to this 30 one is the opposite side all right so one is opposite and in relation to this 30 this side is adjacent and this side over here opposite from the right angle is the hypotenuse let's use that to write it down for the 30° angle so the sign of 30 would be opposite over hypotenuse which gives us 1/ 2 the coine of this angle here is the adjacent side divid the hypotenuse which is < tk3 or < TK 3 / 2 and the tangent of this angle is the opposite side ID the adjacent side which is 1 over the square Ro TK of 3 let's look at the 60° Angle now for the 60° angle this side is right across from it so this side is the opposite to this angle this one is still the hypotenuse so the hypotenuse doesn't change but the opposite will change depending on which angle you're talking about and adjacent will change depending on which angle you're talking about hypotenuse never changes so this angle so this side now becomes the opposite to it and therefore this side becomes the adjacent side to it so for 6 the S of 60 is opposite over hypotenuse which is < tk3 ID 2 and the cosine of this 60° angle is the adjacent side over the hypotenuse which is 1/ 2 and the tangent is the opposite side divided by the adjacent side which is root3 so you can see that there's a little relationship thing going on here between two of the numbers you can see that the S of 30 is equal to the cosine of 60° both of them are equal to a half you should also see that the cosine of 30 is equal to the S of 60 and both of those are equal to the < TK of 3/ 2 the tangent of 30 and the tangent of 60 are reciprocal to each other but I'm primarily interested in these two what do we notice about 60 and 30 well 60° + 30° is is equal to 90° and these angles when two angles add to give 90° we call them complementary angles so if two angles are complementary that is if a whatever angle that is plus b de is equal to 90° then s of a is equal to cosine of b or you can turn it around cosine of a is equal to S of B this is a very important relationship to remember in any triangle as long as the two angles add to give you 90° that is they are complementary then the S of a will always be equal to the cosine of b or vice versa let's look at another one here we have a isos triangle so let this side be one and this side be one then we need to find the length of this missing side to find the length of this missing side we can use Pythagoras's Theorem let's do that first remember Pythagoras's Theorem states that the two shorter sides when you square them and add them you get the length of the longest side let's call this side x for now so one square let's write down Pythagoras's Theorem first it would say something like a 2 + b sare is = to c² so in this scenario 1 2 + 1 2 gives you x² where X is our hypotenuse 1 square is 1 what 1 square is 1 so 1 + 1 is 2 is equal to the root of x² is equal to X3 rather and um finding the square root X is therefore equal to the square < TK of 2 so we can write that here that X is equal to square < TK of 2 so this side has a length of < tk2 or Square < tk2 for our purposes I'm not going to find that number in a calculator because I want to keep it exact once you put this number in a calculator it's an IR rational number you will not be able to contain it so let's keep it as < tk2 now what is the sign remember s is opposite over hypotenuse so we have s is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is opposite over adjacent which gives you your sua TOA idea that helps you remember the ratios so for the S of 45 sign is opposite take this 45 here this is opposite this side would be adjacent and this one is hypotenuse s is opposite over hypotenuse so 1 / < tk2 cosine of angle is one it's adjacent over hypotenuse which is the same thing as 1/ < tk2 and the tangent of 45 is opposite over adjacent which is 1 over 1 which is one so these three are also very important to remember so in any triangle any any any isos triangle with a 45 like that then cosine of a is equal to S of B if state that very clearly if a equal B and here both sin 45 and cos 45 have the same answer so cos 45 is equal to sin 45 which is equal to 1 over < TK of 2 and very importantly tangent of 45 is equal to 1 these little things make a big difference when you're working out multiple choice questions so tan 45 isal to 1 very important all right we're now going to look at how to find two other ratios given one ratio now remember we said that there are six ratios but for CX purposes we concentrate on only three s cosine and tangent and here we say if the sign of the angle is equal to 11/ 10 then we can find the other two the other two ratios related to that one so in order to do this the first thing we do is that we draw a triangle so let's draw a triangle here and let's put in an angle now the sign of this angle is equal to 10/ 11 sign as you remember s is opposite over hypotenuse cosine is adjacent over hypotenuse and tangent is opposite over adjacent so the sign of the is opposite over hypotenuse which means this is the opposite side this is the hypotenuse this is the adjacent side it's 10 over 11 so this side is 10 this side is 11 and what we want to find is the adjacent side now by pythor's theorem this side Square let's call it X x² plus this side Square 10 s supposed to give us 11 Square we know that the 10 must go here because the 10 is opposite and 11 is the hypotenuse so we put it there so x² + 100 is = 121 which means that x² is equal [Music] to 21 and X is therefore equal to the square root of 21 again we're going to keep that as it is um and without finding the the the the decimal answer now that we have this we can write down the other two ratios so f sign of this angle um is this so the cosine of that angle is going to be equal to the adjacent side over the hypotenuse which is < TK 21 over 11 and the tangent of that angle is going to be equal to the opposite side divided by the adjacent side which is 10 over theun of 21 so if a if a CX equ question gives you one ratio say sin 5 is equal to 11 10 what is the value of cos or tan F then you can actually just you can actually find the number now the this angle here have another one let's practice it again the tangent of the angle here is equal to 2 let's draw a triangle that's how we start them tangent is equal to opposite over adjacent opposite over adjacent put in an angle then this is the opposite side which means that this has to be two and this has to be one because for us to get two we have to have two / one now we can use Pythagoras to find this side by Pythagoras's Theorem this is the hypotenuse 1 2 + 2 s would give us x s so we have 1 + 4 gives us x² which means that X is five square root of five big your par so having found that we can then for we can then write down the S and the cosine so the sign of that angle is going to be equal to opposite over hypotenuse which is 2 over < TK 5 this side is equal to the < TK of 5 and the cosine of the angle is equal to the adjacent side divided by the hypotenuse which isun 5 here we have another situation where we have the cosine of an angle is equal to 0.7 so first we're going to just change this 0.7 to a fraction this is the same thing as 7/ 10 and so putting that on a triangle remember cosine cosine is adjacent divided by hypotenuse which means that the adjacent is here so this is seven and this is 10 and so now we we need to find this side using Pythagoras's Theorem again x² + 7 2 2 shorter sides squared is equal to the longest Side Square so we have x² + 49 is = to 100 x² is therefore equal to 100 - 49 and x² there so is equal to 51 we're not going to find the root of it we're going to leave it as is so X is equal to the equal to theare root of 51 not going to put this in a calculator leave it exactly like that with the root sign in there so this sign is equal to the this side is equal to the square root of 51 and know that you have that we can write down what are the other two part from C the other two are s so the sign of the angle will be equal to opposite over hypotenuse which isun 51 over 10 and the tangent of the angle is equal to the opposite which is RO otk 51 over adjacent side which is seven these situations come up a lot in multiple choice papers like I said and it makes sense to know them it makes sense to become familiar with them