in this lesson we're going to talk about the law of sines so let's go over the formula so let's say if we have a triangle where this is angle a b and angle c across angle a you have side a across angle b there is side b and across angle c is side c so the letters that are capitalized represents the angles the lowercase letters represents the side lengths and here's the equation that corresponds to this triangle side a over sine a is equal to b over sine b which equals c over sine c so you could use this equation which is known as the law of sines to find any missing angles or sides in a triangle and if you recall the three angles of a triangle must always add up to 180 degrees any three-sided figure will always add to 180. so let's work on an example let's say if we're given angle a which is 60 degrees and we're given angle b which is 70 and also side a let's say side a has a length of 8. go ahead and solve the triangle so you want to find everything that's missing you want to calculate angle c side b and side c the first thing i would do is draw a picture so let's label this as angle a b and c so we have the value of angle a it's 60 degrees and angle b is 70. to find angle c we know that a plus b plus c has to add up to 180 so a is 60 b is 70. let's find the missing value 60 plus 70 is 130 and 180 minus 130 is 50. so the missing angle is 50 degrees so that's angle c now we could use the law of sines to find the missing two sides okay what just happened here side a is 8. let's solve b so we're going to use this portion of the law of sines sine b actually rather b over sine b is equal to a over sine a the law of sines have three fractions but you only need to use two out of those three fractions at any given moment now let's go ahead and plug in what we have we don't have the value of b but we know that angle b is 70 degrees side a is 8 and angle a is 60. now in order to find the value of b we need to cross multiply so this is going to be 8 sine 70 which is equal to b times sine of 60. now in order to isolate b by itself let's divide both sides by sine 60. so this is what you want to type in in your calculator type in 8 times sine of 70 degrees and then divide that by sine 60. if you don't get the right answer perhaps your calculator is in radian mode so make sure to change it to degree mode so you should get 8.68 so that's the value of b now let's go ahead and calculate the value of c so we're going to use this equation a over sine a is equal to c divided by sine c so we don't need b in this example side a is 8 divided by sine of angle a which is 60. we're looking for c and angle c is 50. so once again let's cross multiply so this is going to be 8 times sine of 50 which is equal to c sine 60. so now let's divide both sides by sine sixty so c is going to be 8 times sine of 50 degrees divided by sine of 60 degrees so c is equal to 7.07 and that's it so that's how you can find everything in this triangle and that's how you can use the law of sines to find the missing angles and missing sides now let's try another example so let's say angle a is 42 degrees side a is 10 and side b is 9. so feel free to pause the video and solve the triangle so once again let's draw the picture first so here we have angle a b and c angle a is 42 degrees actually i should put that here side a is ten side b is across angle b so that's nine so what we have is a side side angle triangle so it's an ssa triangle let's start by finding angle b so we're going to use this formula a over sine a is equal to b over side i mean sine b a is 10 angle a is 42. side b is 9 and our goal is to find angle b so let's cross multiply so we're going to have is 9 times sine 42 which i'm just going to go ahead and get the decimal value of that's about 6.022 and that's equal to 10 times sine of angle b so now let's divide both sides by 10. so 0.602 is equal to sine of b now in order to find angle b you need to take the inverse sine of 0.602 so arc sine of that answer will give you an answer that's 37.03 degrees now sometimes you may have more than one solution so anytime you're using the inverse sine function you need to keep in mind you may get two answers the second answer angle b could be 180 minus 37.03 and let's just round it to 37 just to keep things simple because 37.03 that's very close to 37. so 180 minus 37 that's 143 now here's a question for you are we going to have one triangle or two triangles in this problem is there one solution or two solutions the first solution will always work but what about the second solution is this angle possible it turns out that it's not if you add angle a and angle b 42 plus 143 that's 185 that exceeds the maximum angle that could be inside a triangle which is 180. the sum of all three angles and the triangle has to be 180. so if these two angles exceed or if it's equal to 180 it won't work it has to be less than 180. so because those two angles add up to a value that's greater than 180 we're only going to have one triangle instead of two triangles so we can get rid of the second solution so angle b is 37 degrees now let's go ahead and find angle c so we know that a plus b plus c is 180 so to find angle c is just going to be 180 minus the other two angles so it's 180 minus 37 well a is 42. so there's gonna be 180 minus 42 minus 37. 180 minus 42 is 138 and 138 minus 37 that's 101 so that's the value of angle c so now we could find the last missing side side c so let's use this formula a divided by sine a is equal to c over sine c side a we know it's 10 angle a is 42. side c we're looking for that angle c is 101. so let's cross multiply 10 times sine of 101 degrees that's 9.8163 and that's equal to c times sine 42 so side c is going to be 9.8163 divided by sine 42 which is 14.67 sometimes it's good just to glance at your answers to see if it makes sense the smallest or the shortest side is going to be across the shortest angle side b is the shortest side and notice that it's across the lowest angle which is 37 side c is the longest side and it's across the longest angle so just by looking at that you can tell if your answers make sense or not so if you see a very large angle associated or across a very small side you know something is wrong so that's just a quick way to see if you have the right answer or if your answer makes sense here's another example of an ssa triangle let's say that angle a is 75 degrees side a is eight and let's say that side c is nine go ahead and solve the triangle so let's start with a picture as usual we're going to call this angle a b and c so angle a is 75 degrees side a is eight side c is nine so let's start by finding angle c so let's use this formula side c over angle c or sine of angle c is equal to side a over sine of angle a c is nine a is eight and angle a is seventy five so let's cross multiply as usual 9 times sine of 75 that's 8.693 and that's equal to 8 times sine of angle c now let's divide both sides by 8. so 8.693 divided by 8 that's about 1.00 and that's equal to sine of c so to find angle c we need to take the arc sine or the inverse sine of 1.089 now if you type in arcsine of 1.089 in your calculator it will give you an error sine has the limited range it's between negative 1 and 1. so if you try to take the arc sine of a number that's larger than one it's not going to work what this means is that this triangle has no solution we can't solve it so there's nothing we could do with this problem let's try this problem let's say that angle a is 30 degrees side a is 7 and let's say we're given side b which is 8. take a minute and work on this example always start with a picture so angle a is 30 degrees side a is 7 and b is 8. so the first angle we need to find is angle b so let's use this equation a over sine a is equal to b divided by sine of b so a is 7 and capital a is 30. b is 8. we need to find angle b so let's cross multiply eight times sine of thirty sine thirty is one half times eight that's four and that's going to equal seven sine b so four divided by seven is equal to sine b so therefore angle b is equal to the arc sine of 4 over 7. and so that's going to give you about 34.85 degrees so that's angle b let's round that to 34.9 now we need to see if we could get a second solution or a second triangle so we're going to subtract 180 by 34.9 so that's 145.1 now if we add this new angle with the pre-existing angle notice that it's less than 180 30 plus 145.1 is 175.1 so therefore we can get two possible solutions whenever you're about to get two possible solutions it might be wise to draw a second triangle so the pre-existing values will remain the same a is going to still be 30 and side a is 7 side b is 8 but angle b now is 145.1 so that's the difference now let's get rid of a few things so let's calculate angle c in the first triangle so that's going to be 180 minus 30 minus 34.9 so let's subtract those values so you should get 115.1 and now for the second triangle it's going to be 180 minus 30 minus 145.1 so this is going to be 4.9 degrees now we got to find side c for both triangles and then we'll look at our answers to see if it makes sense so let's use this formula c over sine c is equal to a over sine a c we don't have the value of c we need to find it angle c is 115.1 in the first example a is 7 and angle a is 30. so what we need to do is we need to multiply 7 by sine of 115.1 and that's going to equal c over sine 30. so to get c by itself we need to divide both sides by sine 30. so i'm just going to go ahead and do that now so you should get 12.7 for side c now let's do the same thing for the second triangle so this time angle c is 4.9 degrees a is still seven so this is going to be seven times sine of four point nine which is equal to c times sine of thirty and then we're going to divide both sides by sine thirty so seven times sine four point nine divided by sine of thirty that's going to give you one point i guess you could round it to one point twenty or just one point two now looking at the first triangle doesn't make sense notice that the shortest side is across the lowest angle which is 30. the longest side is across the largest angle which is 115.1 so that makes sense looking at the second triangle 1.2 is the shortest side which is across the smallest angle eight is the longest side and that's across the largest angle so both triangles make sense and that's how you could solve it so now you know how to solve a triangle when there's two solutions so anytime you're solving for an angle like we got 34.9 for angle b find the second angle find an angle that's supplementary to it by taking 180 and subtracting by 34.9 that gave us 145.1 and then once you get that second angle add it to the pre-existing angle if the sum is less than 180 then it's possible that you can have two triangles if the sum is greater than 180 then there's only one solution only one triangle can be formed you