[Music] in this video we're going to learn about the sign rule let's start by taking a triangle and we're going to label its sides a b and c now we're going to label the angles inside this triangle and we're going to label each of the angles with the same letter as the side that's opposite it so opposite the side which is labeled a we're going to put an angle or a as well but we'll use a capital letter so that we know that the angles are capitals and the sides are lowercase opposite the side B we have angle B and again we use a capital and then opposite the side C we have capital c for this Angle now it turns out that if you pick any of the sides for example a and then we divide this by the sign of the angle opposite so divide this by S of a this will give you exactly the same value as if you did this for any of the other sides so if you took side B and divided it by S of capital B and this also gives you the same value as if you took side C and divided it by the sign of capital c and this here is known as the sign rule you can use the sign rule for any triangle and we're going to use it now to find a missing side in this triangle here now even though the sign rule has the letters a b and c very rarely do you actually need all three of the letters at all in fact in almost all cases you just need to use a and b so to approach this question we first first of all going to add some labels to the diagram we're going to try and label lowercase a capital a lowercase b and capital B it's always good to try and make the one that you're finding the a so since we're trying to find the side which is marked X I'm going to make that lowercase a if this is lowercase a the angle opposite this must be capital A so the 85 must be capital A then I would label this 11 cm on the right as lowercase b and the angle opposite that one as capital B so the 35 is capital B then we write out the first part of the sign rule so a over sin a = b/ sin B and then we're going to write this once more but we're going to substitute in all of the numbers from the question so if we start with lowercase a well that's the side we're trying to find which is X so let's write an X there then we divide this by sign of capital A and capital A is 85 so divide by S of 85 then moving to the top right we've got lowercase b we that as 11 and on the bottom we've got s of capital B and capital B was 35 so s of 35 what we have now is just an equation to solve to try and find X on the left hand side it says x / s of 85 so we can find X by multiplying both sides by S of 85 if you multiply by sin 85 on the left that will cancel the S 85 that's already there so on the left you just have X and on the right hand side you've got 11/ sin 35 but we're going to multiply this by sin 85 you can type this into your calculator as it shows here but you may also write it like this as 11 sin 85 over sin 35 so now you go ahead and type this one into your calculator and read off the value so for this one we get X is equal to 19104 937 the question may give you an indication for how accurate to give your answer often one decimal place or three significant figures so let's round this one to one decimal place and that would be 9 .1 and we also need some units on there and we can see this one was in cenm so 19.1 CM now let's try a second example so in this question here we may be asked to find the length of BC since there's no label here I'm going to label on BC with an X to remind me that that's the one we're trying to find now before we go ahead and label the sides and angles for this triangle I want to point out that the corners of this triangle have actually been labeled using a B and C this is very common in exam questions since it makes it very easy for the paper to signal to you which side angle or point it's referring to however for us in this question it's quite unhelpful since we're actually going to label some of the angles and sides using those letters in a moment so if you do get a question which uses this labeling technique you may want to just cover up those or give them a quick scribble out temporarily so that you don't get confused about which letters are which now we can go ahead and add our labels as we did before we're going to start with labeling lowercase a which is the side we're trying to find so that's where the x is then the angle this opposite this one would be capital A so that's the 80° over here then we label the other side in the question as B so this 30 cm is B and the angle opposite that one is capital B so that's this one here then we can go ahead and take the sign Rule and write that out and then we'll write it out once more but replace all of the letters with the information we've got in the question so instead of lowercase a that's X the one we're trying to find instead of capital A that's 80 so we've got s of 80 instead of lowercase B it's a 30 and instead of capital B it's 93 so we've got s of 93 then as before we're going to multiply both sides by S80 so that we get X on the left hand side so on the left hand side we would have X and on the right hand side we've got 30 over sin 93 but we've just multiplied this by sin 80 and as we did before we may write this as one fraction which is 30 sin 80 over sin 93 type this into your calculator and you'll find that X is equal to 29607 3207 let's round this one to one decimal place once more so this will be 29.6 and if we add some units which are centimeters now you can also use the sign rule to find missing angles in triangles the previous two questions we use were finding missing sides if you are finding a missing angle using the sign rule you want to take the reciprocal of all of these fractions so instead of a over sin a we want sin a over a and the same thing happens to the other fractions too so this is the version of the formula you would use if you were trying to find a missing angle let's have a look at how we can do that in a question so for this question here you can see the angle this time is what's missing and that's been labeled X so we're going to start by labeling that X as a but since it's an angle it will be capital A the side that's opposite this will be lowercase a the other angle given in the question will be our capital B so that's the 88 and the side opposite this will be the lowercase b so 25 we then take the first part of that formula once more and we write this out but replace all of those letters with the information in the question so instead of s of capital A that's now s of X the one we're trying to find we divide this by lowercase a so that's 18 and on the second fraction we've got s of capital B so s of 88 and this is divided by lowercase b so that's 25 to solve this one we're going to multiply both sides by 18 if we multiply the left side by 18 that will cancel the divide by 18 so we've just got s of X on the right hand side we have sin 88 over 25 and we've just multiplied this side by 18 so multiply by 18 as in the previous two questions you might write this as one single fraction so 18 sin 88 over 25 then you can type the right hand side into your calculator and this will give you a decimal on the left hand side though we still have S of X not X but s of X and the right hand side using your calculator will give you this number here now this isn't the answer to the question because this is not the value of x but rather the value of s of x to find the value of x we need to do the inverse sign of both sides if we do the inverse sign of the left hand side this will give us X and if we do the inverse sign of the right hand side we need to type this into our calculator the inverse sign of this number here if you do type that into your calculator you'll find that X is equal to 46.01 8283 let's round this to one decimal place again so that will be 46.0 and since this one's an angle we'll put a degree sign let's try one more of these so once again we're finding an angle so we're going to use the angle version of the sign Ru formula we're going to label the angle we're trying to find Capital a the side opposite this lowercase a the other angle in the question capital B and the side opposite this lowercase b then we take the formula and write this out once more but replace all of the letters with the information from the question so we've got s of a which is s of X over lowercase a which is 5 is going to equal s of capital B which is s of 110 divide by lowercase b which is 12 we then multiply both sides by 5 so on the left hand side this will cancel the five so we've got sin x and on the right hand side we have sin 110 over2 and this is now multiplied by 5 which we could write as 5 sin 110 / 12 the left hand side will remain a sin x and we type the right hand side into the calculator which gives you this number here we then use inverse sign so X will equal the inverse sign of that number there and using your calculator that will give you this number here we can round that off to a suitable degree of accuracy let's go one decimal place again so this one will actually Round Up to 23.1 De and that's the answer to this question now we're going to look at one more example of using the sign R to find an angle but something very peculiar happens in this question we're going to attempt to solve it in the way we did the other questions so we'll Begin by labeling so the one we're trying to find is going to be capital A the side opposite this lowercase a the other angle capital B and the side opposite this lowercase b then we take the formula and we're going to substitute the information from the question so instead of s of capital A at s of X instead of lowercase a it's 20 instead of s of capital B it's s of 33 and instead of lowercase B it's 12 we can multiply both sides by 20 like we did before this will give us sin of x = s of 33 over2 multipli by 20 which is 20 sin 33/2 then we can type this right hand side into the calculator so we get S of x equals this number here then we use inverse sign to work out the size of X so we would have x equals the inverse sign of this number which gives you 65.2 de now at this point you might notice something a little bit off if you look at the diagram of the triangle this angle here doesn't look like 65.2 de at all in fact it looks obtuse it looks greater than 90° so what's happened here well if we have a look at the graph of sin of X let's draw some axes and we know the graph of sin of X goes between 1 and minus one and let's mark on the key points 180 and 360 and the graph of yal sin of X would look something like this now when we end up with an answer of X = 65.2 De this is because if you mark 65.2 de on the x- AIS go up to the graph of s of X and then read across you get that value of 0.977 and so on now if you look closely at the graph there's actually a second value another angle that would give you this value too so if we extend that line horizontally across go to the sign graph and then go down we end up at this angle here so the sign of this angle will also give us 0.977 and so on but what is this angle well due to the symmetry of the sign graph we can find this angle by subtracting the other angle 65.2 from 180 if you do 180 minus 65.2 you get 11 14.8 so the sign of 65.2 and the sign of 11 14.8 both give you 0.977 and so one so in this question it might well be the case that the actual answer is 114.1 but the question would need to give you some information to indicate which of the two angles in it wanted so for example the question may say given X is an obtuse angle now I missed that information out obviously because I didn't want to give away what was happening in this question but the exam question will always give you some indication of whether that angle is obtuse or acute this will help you choose the right one when you come to do your answer so if the question did say given that X is obtuse we would finish this question by taking that angle 65.2 away from 180 getting the true answer of 11 14.8 but it is entirely possible the question could say given that X is an acute angle in which case the answer would have been the original one we found of 65.2 De however there's a new problem now the diagram doesn't look like that 65.2 de it looks like an obtuse angle so what's happening here well it turns out there are two possible ways of drawing this triangle what we can do is move this 12 CM across like this and extend that bottom line and we haven't actually changed any of the information of this triangle at all the sides are still still the same length the 33° is the same the only thing that changed is that angle X so if this were the case the diagram should actually really look like this in which case the original answer would have been 65.2 de. this particular situation has a special name we call it the ambiguous case because unless you're given some further information you can't tell which of the two angles is the correct one so rest assured in your exam question if this does happen you will be given some information for example it will tell you whether the angle is acute or OB use thank you for watching this video I hope you found it useful check out the one I think you should watch next subscribe so you don't miss out on my future videos and now go ahead and try the exam questions I've Linked In this video's description