we use the sign rule to find a missing length or a missing angle in a non-right angled triangle so for right angle triangles we have pythagoras and socatoa but for non-right angle triangles we can use the sine rule or the cosine rule so here we've got the sine rule so we've got a over sine a equals b over sine b so the little a's are lengths so if this was little a the big a's are the angles and little a is opposite big a so this is little a opposite it is big a if this length is little b opposite it is big b so sine rule is for opposites there are two versions of the formula of the sine rule we use lengths on top to calculate lengths so a over sine a equals b over sine b for lengths and for angles we just flip it over and put the angles on top sine a over a equals sine b over b we do that to make the working out easier so let's have a look at an example this question says find the value of x the first thing i'm going to look for is opposites do i have opposites so can i use the sign rule so i've got x opposite 85 and i've got 12 opposite 53 so i can use the sign rule i'm working out a length so i'm going to use the same rule for lengths so i'm going to put the lengths on top a over sine a equals b over sine b so the little a's are the lengths the big a's are the angles so x is little a 85's big a if 12 is little b 53 is big b and we just need to substitute these in to our sine rule now so little a we're going to call x sine a big a is 85 so x over sine 85 is equal to our little b we're calling 12 and 12's opposite is 53 so 12 over sine 53. the angles always go with the signs we want to get x by itself at the moment x is divided by sine 85 so to get rid of that we multiply both sides by sine 85 so x is equal to 12 over sine 53 times sine 85 and we can just type this exactly how it looks into the calculator to find x so i'm going to write fraction button 12 over sine 53 and then all of that times sine 85 and i've got an answer of 14.968 and so on i will give my answer to two decimal places so that'll be 14.97 so x is 14.97 and it's centimeters to two decimal places okay a question for you to try so pause the video and give it a go okay so firstly do we have opposites so x is opposite 34 degrees 15 is opposite 97 degrees so we can use the sine rule we're working out lengths again so we're using the sine rule for lengths which is a over sine a equals b over sine b so if we have x as little a that means 34 is big a if we have 15 as little b that means 97 is big b so substituting in little a is x so x over sine a's opposite is 34 equals 15 over 15 is opposite which is 97 so be sine 97 to get x by itself i'm going to multiply both sides by sine 34 so x is 15 over sine 97 times sine 34 so we type into the calculator exactly how it looks and again i'm going to give the answer to two decimal places so we've got 8.45 this time and again centimeters and i write 2 db okay another example so this time it says find the length of a b so a b is this length here a to b so i'm going to call that x and i'm looking at my opposites so 8.5 centimeters is opposite 81 so i've got one opposite there but x x is opposite angle i don't know so can i use the sine rule if i don't know this angle i can't but i can work out that angle so the angles in a triangle add up to 180 degrees and we've got 81 and 39 here so we can find the missing angle 81 and 39 that makes 120 and 180 minus 120 is 60. so you can use a calculator so we can work out that 60 degrees and we can use the sine rule because we do have opposites so we're working out a length again so we're going to put the lengths on top a over sine a equals b over sine b and i'm going to label it again so i have x as little a so 60 would be big a and the other length is little b and it's opposite is big b so we substitute in so a over sine a so x over sine 60. equals 8.5 over sine 81. so x over sine with its opposite equals 8.5 over sine with its opposite we want to get x by itself so we're going to multiply both sides by sine 60. and we just type this into the calculator to get our answer so 8.5 over sine 81 times sine 60 and two decimal places again i've got 7.45 still centimeters to two decimal places okay a question for you to try so pause the video give it a go and press play when you're ready to carry on so find the value of bc so we're working out the length b to c and do we have opposites can we use the sign rule so we've got 9.3 opposite 52 we don't know x is opposite just yet but we can work it out from finding the missing angle in a triangle so 180 take away 52 and take away 93 tells us the missing angle is 35 degrees so we do know opposites we've got x and it's opposite is 35 degrees 9.3 is opposite is 52 degrees so we're going to use the same rule for lengths again so it's going to be x over sine with its opposite which is 35 is equal to the other length 9.3 over sine with its opposite which is 52 degrees so multiply both sides by sine 35 so x is 9.3 over sine 52 times sine 35 type it into the calculator and to two decimal places i have 6.77 it rounds up so 6.77 centimeters to two decimal places okay this time we're being asked to work out an angle so the question says find the size of angle abc that means a b c this one here so do we have opposites can we use the same rule our angle x is opposite 5.2 and our other angle 96 is opposite 9.1 we've got opposites we can use the sine rule this time we're going to use the sine rule for angles so it's going to be sine a sine big a over little a equals sine big b over little b so the sign always goes with the angle and it's the big letter is the angle and the little one is the length so this time we're going to have x as big a and opposite it 5.2 will be small a our other angle is going to be big b and its opposite will be little b so let's substitute in so we've got sine x this time over its opposite which is 5.2 equals sine 96 other angle on top angles are on top this time over its opposite which is 9.1 so we're going to get sine x by itself by multiplying both sides by 5.2 so we can have sine x is equal to sine 96 over 9.1 times 5.2 so we'll type that into the calculator and we'll get our answer for sine x we'll know what sine x is so sine 96 over 9.1 times 5.2 so we've got 0.568 and so on so i'll write sine x is equal to 0.568 and so on but that's sine x that isn't x so how do we find x how do we get sine away from x we do the opposite of sine which is shift sine so the inverse of sine shift sign and then we do our answer in there shift sign answer and we get our angle so 34.6 three to two decimal places so x is 34.63 degrees to two decimal places okay a question for you to try so pause the video give it a go and press play when you're ready for the answer okay do we have opposites so we've got our angle x opposite 18 and our angle 57 opposite 25 so we've got opposites so we can use the sign rule if we have x as big a that means 18 it's opposite it's going to be our little a and if we have 57 as big b it's opposite 25 will be little b so we're going to use the sine rule for angles so angles on top so sine always goes with the angle so sine x over its opposite 18 equals sine 57 the other angle with its opposite over its opposite 25 we'll get sine x by itself by multiplying both sides by 18 so sine x equals sine 57 over 25 times 18. so let's type that in the calculator so 0.603 and so on but that's sine x we want x so to get rid of sine to get sine away from x we shift sine so shift sign the answer and that's 37.15 to two decimal places 37.15 degrees and again two decimal places i'm giving my answers to okay one more example of this this time angle c a b is obtuse and that is important so obtuse means between 90 degrees and 180 degrees so the angle is obtuse work out the size of angle c a b c a b so this is x c a b do we have opposites we do so our angle x is opposite 12 our angle 44 is opposite 8.6 so we're working our angles we can use the same rule for angles so sine a over a equals sine b over b working at angles we put the angles on top the angles the big letters the capital letters always go with the sign so we've got sine a which will have a sine x i'll quickly label that so big a little a big b little b so sine x over its opposite which is 12 equals sine the other angle 44 over its opposite which is 8.6 to get sine x by itself multiply both sides by 12. so sine x is sine 44 over 8.6 times 12. so let's type that into the calculator so sine 44 over 8.6 times 12 and we've got our sine x of 0.969 and so on 0.969 and so on but we're going to find x so to get sign away from x we're going to do shift sign so if we shift sign the answer we get an answer of 75.76 so i've got 75.76 i'll write that down 75.76 degrees but hold up that's acute we want to work out an obtuse angle this time and we've been given an answer that is less than 90. so the calculator's given us an answer but it's not the answer we want so we want an obtuse answer an answer between 19 180 and the calculator has given us 75.76 so to find our correct answer we're going to do 180 take away the answer we were given and get 104.24 so x is going to be 104.24 degrees and i will explain how that works quickly so if we've got a sine graph which looks something like this so we've got sine x and i'll write the y axis and the sine graph is a wave and it's a wave and it goes on forever and all right 90 180 270 360. and we had 0.969 so we said to the calculator i've got 0.969 tell me what angle that is and it said the first angle the calculator always gives us the first angle on the graph which is 75.76 and usually that's okay we're happy with the first one but in this case we actually wanted the second one and the calculator doesn't give us that answer so we have to do an extra calculation and using this symmetry so saying the same gap here so this gap here which was 75.76 is the same as this gap here we can work out the second answer by doing 180 take away 75.76 to find the second answer so when we are finding an obtuse angle with the sign rule we do 180 take away the answer the calculator gives us to find the obtuse angle okay let me clear this and here's a question for you to try so pause the video give this one a go and press play when you're ready for the answer so this question says find the possible values of x so there's going to be two possible answers the diagram is not drawn to scale so there could be an acute answer and an obtuse answer and let's just quickly go back and say that that wasn't possible in the previous two questions because in this first question here if we'd done 180 minus 34.63 we would have had an angle of 145 degrees and that wouldn't work because that would go over 180 and again for the second one if we had an answer of 143 degrees that would be over 180 so there's not two possible answers so back to the question find the possible values of x so firstly can we use the sine rule do we have opposites and the answer is yes we're working on angle so we're going to use the same rule for angles so angles on top sine a over a equals sine b over b we're going to call our x big a and opposite will be little a our other angle is big b opposite it is little b so sine x over its opposite which is 5.2 equals sign with the other angle 31 over its opposite which is 2.7 so sine x multiplying both sides by 5.2 is sine 31 over 2.7 times 5.2 so we're going to type it into the calculator sine 31 over 2.7 times 5.2 so we've got 0.991 and so on that sine x to get x by itself i'm going to shift sign so shift sign my answer and that's 82.71 to two decimal places degrees that's one possible answer that's the acute answer there's also an obtuse answer so 180 take away the answer and that's 97.29 so it could also be 97.29 degrees okay to finish up two questions pause the video give them a go and press play when you're ready for the answers one find the length of bc so this is going to be x b to c do we have opposites x is opposite 96 9.1 is opposite 57 we're working at a length so let's use the sine rule for length with the lengths on top so a over sine a equals b over sine b so we have x as little a opposite it will be big a the other length little b opposite it will be big b so we've got x over sine with 96 the other length 9.1 over the sine of its opposite which is 57 and to get x by itself we're going to multiply both sides by sine 96 and then we're going to type it into the calculator so we've got 9.1 over sine 57 times sine 96 type into the calculator and we get 10.79 to two decimal places so x is 10.79 centimeters to two decimal places question two find the value of x this time we're working at an angle do we have opposites yes we do so we're going to work out an angle so let's put the angles on top so that big a and little a big b and little b so sine sine always with the angles so sine a sine x over 3.5 equals sine with the other angle 99 over 4.4 we want sine x by itself so let's multiply both sides by 3.5 so we can type that into the calculator to get our sine x so sine 99 over 4.4 equal no sine 99 over 4.4 times 3.5 is 0.785 and so on and we want to get x so to get sine away from x to get x away from sine we shift sine so we're gonna shift sign shift sign our answer and we get 51.78 to two decimal places so 51.78 meters to two decimal places you