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0:01 [Music] in this video we will discuss the law of science but before we proceed to our discussion please don't forget to like our video and subscribe narinkayo if we will recall the topics that we have discussed in the previous weeks uh we talked about the angle of elevation and angle of depression where it focus on right triangles okay so when it comes to the law of science we already have oblique triangles so when we are dealing with oblique triangles ibxabihen right angle so unlike in our previous
0:53 discussions previous lessons we focus on the right triangles now this time we will focus on oblique triangles this means that there is no right angle in the given triangle so in order to solve the triangle we need to know the three parts of which at least one is aside so marin tying three parts detox oblique triangle adding sides your small letters this is represented as small letter a b c at the point adding sides now at opponent capital letters this represents our angels
1:45 so this is your angle a and the opposite side a which is small letter angle b and its opposite side small letter b and then angle c and its opposite side small letter c so take note of the representations of our sides and angles for sides we are using small letters and for the angles we are using capital letters now let's talk about the law of science so these are the three parts of the law of sines so make sure that you have uh remember all the uh parts of the given law if the given triangle is an oblique triangle so we have here sine a over a so at the point angle this is your side is equal to sine b over b your angle over your side which is also equal to sine c over c so this is your angle over its opposite side okay now take note of the following the law of science can also be written in the reciprocal form so when we are trying to solve oblique triangles using the law of sines you can also make use of the reciprocal of its three parts so puederin
3:04 a over sine e is equal to b over sine b which is also equal to c over sine c so you can make use of this or this one another the law of sines can also take several forms so atom forms none of a form dito is when we are trying to cross multiply in a pokha pagma sol na tayorito so let's say for example so if i will be using a over sine a is equal to b over sine b and then a is the unknown so i have to cross multiply right so that will become a is equal to b times sine a over sine b okay so a is equal to b times sine a over a sine b
4:12 or capacity as the unknown or c as the unknown so six forms a law of sines now when the two angles of a triangle are given with one side we first find the third angle and then use the law of sines to find the other two sides so ato young tip uh given i think giving you two angles let us first find the third angle and then let's proceed to the law of sines and the number four the law of sines can be used in solving a triangle when we know any of the following so we
5:46 so as long as marriage on one side and two angles we can make use of the law of science another is if you have two sides and an angle opposite this sides sides and an angle opposite of these sides a big essence let's proceed so i have here triangle abc with sides a b and c nail in this triangle and given lampona then is opposite angle b so it upon side b which is 15.8 inches so we are asked to find the side of a and c so before we find the measures of its side sabinga
7:01 so given the two sides and the included side so we can uh at a point in the one angles and an included side so solve the triangle shown in the figure so first find nathan munna young third angle so in finding the measure of the third angle given two sides so apply like nothing in triangle some theorem that the sum of the interior angles of any triangle is is always 180 so that's why we are subtracting the sum of angle a and c from 180 k and sample lag interior angle triangle is always 180 so 180 degrees minus the sum of angle a and angle c so that is 180 minus 43.8 degrees plus 62.2 degrees so admonish so we have 180 degrees minus 106 degrees your angle b will be 74 degrees now this time a side a and side c so this is the time where we are going to apply the law of science so gaga meeting now since three parts reports
8:50 is equal to b over sine b bucket okay so let us substitute we have our a and then sine e or sine e is 43.8 degrees is equal to your b which is 15.8 so again small letter pune at n that is the your side and then sine b this is an angle so 74 sine 74 degrees so we have by cross multiplication at the point of the sixth form s since we are looking for a which is the this is our unknown and cross multiply so we have we have a is equal to cross not n so we have 15.8 times sine 43.8 degrees
10:13 [Music] and then over sine 74 degrees okay let us solve it using calculator so first is you have to press the fraction icon and then sign 43.8 0.8 and then close times 15.8 and then close and then arrow down sign 74 and then close so equals net n so that is eleven point thirty eight so ion so your a is equal to eleven point thirty eight inches next for us to solve for c
11:40 62.2 degrees is equal to your b your b is 15.8 and then sine b is 74 degrees so sine 74 degrees so let us cross multiply so we have c is equal to sine 62.2 degrees times 15.8 over sine 74 degrees that is all so press the fraction bar sine 62.2 degrees and then close times 15.8 and then close over arrow down sine 74 degrees and then close so we have fourteen point fifty four round of nine to the nearest hundred so that is fifth fourteen point sorry fifty four so our c is 15.54 inches
12:47 next so we have here given two angles and one side so again given two angles nothing so given you two angles and one side angagamite not in law of science so since given two angles nothing we can easily find the measure of the third angle angles so we have 180 minus a plus b the measure of your angle a and angle b which is 45 and 50. so that is 95 so 180 minus 95 that is 85 therefore your angle c is 85 degrees so we can now find the measures the length of side b and c so how are we going to find b and c so since marinating given a completo the angle and its opposite side so you know so find let us find first b
13:52 over sine a since so let us substitute b over sine 50 degrees your b is 50 degrees is equal to your a which is 8 over sine 45 so your sine a is 45 so we have to cross multiply since b is our unknown so b is equal to sine 50 degrees times 8 over 40 sine 45 degrees so using the calculator so press the fraction icon or button and then sign sign 50 and then close times 8 and then close arrow down sign 45 and then close so that is equal to 8.67 so that is equal to 8.67 therefore
14:51 your b is equal to 8.67 next let us solve for c so applying again the law of sine since we have two angles and one side so see because is 85 degrees so sine 85 is equal to your a is 8 over sine 45 degrees so cross multiply not n so we have c is equal to sine 85 degrees times 8 over sine 45 degrees let us solve using calculator so press the fraction icon and then sign 85 degrees and then close and then multiply by 8 close arrow down that is sine 45 degrees so close equals so that is 11.27 so your c side c is equal to eleven point twenty seven so again up not in your sides and angles using the
16:15 law of signs now let's talk about the ambiguous case and the by eighteenth
17:12 so there are three possible situations possible and then there is really one triangle that exists and then the other one is there are two distinct triangles okay so kapag ambiguous case again in this case there is exactly one triangle and there are two distinct triangles
18:10 for us to easily identify or determine if there is a solution there is one solution or two solutions okay so let us first discuss the first condition so young first condition has an etopoe if you're a yours a angle a is greater than 0 degrees but less than 90 degrees so in this condition
19:09 the cases under this condition wherein you have an acute angle here because greater than zero but less than 90. so in first case not in case number one if your a is less than b times sine a then your sign b is greater than one so ito i greater than one masmatasa one so
20:00 so there is no triangle form and there is no solution next so a tournament in case number two that if your a is equal to b times sine a then sine b is equal to one so in this case the man kapaging sine b punitin is equal to one exactly one don't know this means that your b atom b not n is a right angle so e big sub meron right triangle once casino marine time right angle automatic there is a right triangle so that is your case number two and then for our case number three if your a is greater than b times sine a and a is less than b
21:52 two angles in a form therefore marine time two triangles then we have two solutions number three so that is case number three and then another last case for condition number one so if a is greater than b times sine a and a is greater than or equal to b so na your a here can be greater than or equal to b so hapaganito in case more than there is exactly one angle that determined so hapaguan anger lung ebiksa b and meron come one solution so ebik sabine meron one triangle next condition if your a is greater than 90 but less than 180 so this is already if you have one object angle says
22:50 angle so in this case this is case number five if your a is less than or equal to b then it can be seen from the figure it open if you will try to look at the figure your a not and mass m x is a b or it can be equal then as you can see there is no solution okay so if your key if your triangle has one obtuse angle and then your a or b i then there is no solution that is for case number five and then for case number six if your a is greater than b
23:45 number letter b so in condition letter b one obvious greater than b then there is exactly one triangle form hence there is one exactly one solution so let's have an example so applying this cases solve triangle abc using the given information your a is equal to 12 centimeters b is 23 centimeters and then your angle a is 34 degrees so find b using the law of sine so we are asked to find b so an unanimous applying the law of science hanapi not in cb since meru nama natalyan given
24:46 is equal to your a which is 12 and sine a which is 34 so sine 34 degrees now we have sine b cross multiply so sine b is equal to sine 34 degrees times 23 over 12. okay so it solved not and shy using calculator we have sine 34 degrees and then close but nothing sorry okay so we have sine 34 and then close times 23 lows and then arrow down we have 12 okay equals so we have one point zero seven two so your sign b is equal to one point zero seven two now if we will go back
25:48 don't samana k says nothing this is under case number one that if your a at a young a not m a is less than b so 12 is less than 23 so a is less than b sine a then sine b is greater than one as you can see your sign b here and i greater than one which is one point zero seven two so e big sub hen b duns in case number one this means that no angle b is determined ebig sabihin there is no triangle form and there is no solution so impossible it is the equation sine be 1.072 has no solution then there is no triangle axis with the given measures impossible yeah satisfy shadows at
27:05 next so given two sides and one angle so again ambiguous case [Music] two sides and one angle solve the triangle shown in the figure so we have solve for b and c b and c and one and a ten so parama not in c b and c we can still apply the law of sine so since my given amanda ditona angle a and its opposite side so applying that in sha we have your b is 11 so we are looking for b eleven i mean sign b so 11 is your b over sine b i think unknown your a is 18 opposition angle a
27:54 so 18 over sine 73 degrees so upon sine b cross multiply so sine b is equal to sine 73 degrees times 11 over 18 so let us solve so we have sine 73 cross i am in close and then l times 11 arrow down we have 18 and then equals so we have 0.58 so 0.5844 now this is under case number six why sabasa case number six if your sign b is greater than zero but less than one so as you can see
28:52 zero but less than one and if your a is greater than b completing the nothing sufficient your a inside a mo is greater than the side b 18 is greater than 11 then there is exactly one triangle form triangle of a form therefore there is exactly one solution okay so ebik sabin continues [Music]
29:38 if so we have that is your sign b or your angle b is 35.70 degrees so eto puyong attenbi 35.76 degrees measures example number one ambiguous key statin
30:25 so 180 minus your e is 73 your b is 35.76 so that is 180 minus 108.76 that is 71.24 so your angle c is 71.24 degrees now applying the law of sines over sine your c is 71.24 degrees is equal to your a which is 18 over your sine a is 73 degrees again
31:24 now let us solve so by cross multiplication we have c is equal to sine 71.24 degrees times 18 over sine 73 degrees so let us solve using calculator fraction bar i mean fraction icon and then sine 71.24 and then close times 18 and then arrow down we have sine 73 and then close so that is 17 point 82 centimeters so a big cib hand our sine c is i mean c side c is equal to 17.82 centimeters next so in this case of triangle abc using the given information adding triangle so i have here b which
32:23 is 28 degrees your b opposite side is 16 centimeters and then your ca which is equal to 20 centimeters which is the opposite of angle c now as you can see let us try to compare the values of b and c sine b so compare nothing is it greater than or less than so young b na then i sixteen so substitute nothing and then your c is twenty sign b your sign b is twenty 28 so it computes 20 sign 28 20 sign 28 okay so i'm lumabus for 9.39 so 9.39 so pinapa compares time sign b so on the mobius podium
33:19 9.39 compared to b which is 16. so therefore 16 is greater than uh 9.39 right so b is greater than c times sine b so that's why unsymbol not in data is greater than so b is greater than c times sine b since b is an acute angle and c is greater than b thinking even is an acute angle because it is greater than 0 but less than 90 and then your c is greater than 16 then this is case this is under case number three this is an ambiguous case and there are two possible solutions number three solutions
34:18 so let us salt so we have to look for the measure of c okay i mean oh yeah four c so we have 16 your b is 16 years b sign b is 28 and then your c is 20
35:34 a bigger triangle sine c sub 1 is equal to sine 28 degrees times 20 over 16 so solve nothing so we have sine 28 degrees times 20 ayan arrow down 16. anglo mabas po is zero point fifty eight sixty eight so ebik sabihin ano pong ebik sabihinito your c sub one s [Music]
36:33 5868 and then close so angladabaspo i 35.93 degrees 35.93 degrees a bigger triangle now then it's a bigger triangle angle a you have to subtract the sum of b angle b and your c angle c sub one so add nathaniel so that is 28 plus 35.93 that is 63.93 therefore your angle a is equal to one hundred sixteen point zero seven at a point so your angle a at a point is one hundred sixteen point zero seven okay
37:48 [Music] so we can apply the law of sines so b since so we can make use of b or c s opposite angle i sine 116.07 degrees so cross multiplication a times sine 116.07 times 16 over sine 28 degrees so nothing so we have sine 116.07 close times 16 so close arrow down sign 28 28 and then close so that is 30.61 so your a is equal to 30.61 centimeters so
39:26 28 at the point i think 16 and then atom that is a smaller triangle so
40:20 their opposite base angles are also equal so since pareho 716 ebooks have been equaled in point opposite angles so this is 180 minus 35.93 that is 144.07 degrees so it'll enhance a smaller triangle so 144.07 degrees
41:26 so 180 minus the sum of 28 plus 144.07 that is 172.07 degrees so 180 minus 172.07 that is 7.93 degrees is okay [Music]
42:20 that is 16 substitute nothing 16 over 28 degrees sine 28 degrees is equal to your a your unknown over your opposite angle is 7.93 so sine 7.93 degrees cross multiply sine 7.93 degrees times 16 over 28 degrees so let us solve that is sine 7.93 and then close times 16 arrow down that is sine 28 and then close so that is 4.7 d so we have our side a or the length of side a which is four point seventy centimeters so nahohan aponatin so as you can see maritime the
43:19 one solution since we have two triangles thank you for watching this video i hope you learned something don't forget to like subscribe and hit the notification bell for updated ko for more video tutorials this is your guide in learning your math lessons your walmart channel
