well hello 3c and welcome to finding sides with primary trig ratios this shouldn't be anything new we did this last year but we're gonna cover it again just to make sure you understand what's going on so you take a look up here at our topic it says finding side links with trigonometry and our goal I can use the three primary trig ratios to solve for side lengths in right triangles so the first thing is what are the three primary trig ratios well we're going to review that here we've got these are the three sine cosine and tangent and I'm just gonna go back over with what they mean what they mean is anytime you have a right triangle like I have here I've got two right triangles actually that if I take if if this theta it has a value let's say it's 50 or something like that any triangle that has a 50 down here would have a 40 up here and if there's a 50 down here 40 up here and 90 here any triangle that has that in it even if it's a great big triangle or little tiny triangle they're all similar which means that if you have similar triangles the side lengths are proportional so anytime I have an angle here I know that when I divide this sides I get something that is gonna be proportional no matter how big the triangle is and your calculator stores all those proportionalities in it so we have to know what the ratios are so I'm gonna click on these things sine is the opposite over hypotenuse cosine is the adjacent over the hypotenuse and tangent is the opposite over adjacent now what do we mean by all of these opposite and adjacent and blah-blah-blah-blah-blah well the first thing you have to know is that we always go by the angle so down here I've got angle theta marked and this theta should've been up here because I wanted it in the opposite way so in this one we've got theta down here and this one we got fade up here and I'm going to erase these numbers because we don't need the numbers we're just going to go by the angles so we go by this angle these three sides are going to be labeled either opposite adjacent or hypotenuse now here's how you tell the difference first of all as we learned with Pythagorean theorem this points the way to the hypotenuse so this side is the hypotenuse always always always always if you've got a right angle it points the to the side that's the hypotenuse which means that we can actually do that one in here too that is the hypotenuse now from this angle I'm going to label the other two sides so if I go directly across the triangle from the angle that I'm working with the one that I circled this side is called opposite because it's directly across which means if I go over into this triangle this is the angle that I marked that's where I put my theta so the opposite side here even though these two triangles are actually identical I just cloned them okay this side over here is called opposite because I'm talking about a different angle so we always go by the relationship with the angle and since the only thing that's left is adjacent we're going to call this adjacent now adjacent actually means next to so this side is actually next to the angle that we're talking about and so up here this is adjacent because this side here is next to the angle we're talking about now the hypotenuse is also next to the angle we're talking about but since it's the special side it gets a special name and we always call it the hypotenuse so now we're going to use these three ratios to find a missing side length and we're going to set them up as a ratio so we need to know how to solve ratios the same way that we were doing in the first lesson from this unit so we're gonna find the missing side length now there's two missing side lengths here there's this one then there's this one so let's actually find them both I'm going to start by finding this one and I'm gonna put a little B on it and they're gonna put the little B on it because it's across from the big B and this side length over here I'm gonna call little a because it's a cross from big a now we're gonna pick which one we're gonna find first when you're using the trig ratios the first thing you should do is circle your angle circle your angle now that's going to be the angle you use so it has to have a number on it cuz using this angle over here would be perfectly useless to us it does not have a number so I'm going to use this angle here and now after I circle the angle I'm gonna label the sides the same way I did up top label the sides so here's how I label the sides I'm going to do that in green this time from this angle well first of all let's put the hypotenuse B this side over here this is the hypotenuse and I'm not gonna write it all out I'm just gonna use a big capital H for hypotenuse now from this angle if I go directly across the triangle remember I get the opposite side and then the only thing left over is a oh look at that big a goes on Lily okay so we got to figure out which we're going to find first are we gonna find little a or little B well if I'm gonna find little a first okay we have to use the a side okay because that's the side Eleison and then I can't use B if I used B side then I have an A and a B together and when I have two unknowns it doesn't work so I have to use a side that has a number on it so there's the number 12 so we say 52 degrees and I have to use side a and I have to use side Oh now do you remember what the shortcut was for the three primary trig ratios and we have sine opposite over hypotenuse cosine Jason over hypotenuse tan opposite over adjacent we use these letters so and these letters ha and then these letters Toa and so sohcahtoa is what you should have in mind when you're doing these things so I'm just gonna write so ha Toa and these two things that I've circled cuz I'm finding little a and they have to use a number is 8 and o 8 and Oh our toe isn't OA so I'm gonna put the T in front of here in the T is tan now tan is opposite over adjacent so our opposite is 12 and our adjacent was our little a now tan of 52 your calculator knows what that number is it's just a number make sure your calculator says degree or has a big D in it like mine does right here okay now I'm gonna find out what that number is tan of 52 so on my calculator I have to type in 52 and then pressed and some of your calculators you'll have to do it the other way around and I'm gonna keep 4 decimal places this is a nasty decimal but if we keep it to four decimal places that will keep us fairly accurate so one point two seven nine nine so instead of tan of 52 I'm gonna write in one point two seven nine nine I hope that's what I said one point two seven nine nine yep this and this are the same number and then we've got 12 over a now remember when you cross multiplying or solving by comparison or whatever I'd like to do cross multiplying in these if there's no number underneath a number like say you just tell you just have one number that it's not action you can always put it over one so now I'm gonna actually use the cross-multiplying approach on this one and say that this number times a which is gonna give me one point two seven nine nine a is going to equal 12 times 1 which is just 12 and then I have to divide both sides by one point two seven nine nine to get this a by itself one point two seven nine nine so now to get my a I have to do this division so doing that division twelve divided by one point two seven nine nine and that's nine point three eight nine point three eight so that side length is nine point three eight now I'm going to do little B if I choose to do little be okay if I choose to do little B I'm gonna switch colors here I'm gonna go with blue I have to use little B so I'm looking at a side H this time and I also have to use a side that has a number on it and that is still the O side now it's true that we actually do know what a is but since I don't know if I got this one quite right I'm not going to actually use it now we're still using this number that we have here this 52 so you know 52 degrees and I have to figure out whether I'm going to write sine Coast and in front of that 52 degrees so you remember our sohcahtoa we're looking at these two letters O and H which one of these has o NH I can hear you screaming I know you're saying sign has it there we go so so I got to pray sign in front of there and I'm gonna put sign is o divided by H that order remember when I did it up here I'm going to go up very quickly mmm see up here sign opposite over hypotenuse so we have to go opposite over hypotenuse so it's gonna be 12 over B 12 over B now remember sine of 52 the same as tan of 52 is just a number and the calculator is going to tell me what it is so I type in 52 and hit sign and equals so 0.6 oh I missed it zero point six one five six eight six one five six eight and that's 12 over B we're going to put this over one and now we're gonna cross it up we're gonna cross multiply we're gonna go this way and that's going to give me zero point six one five six eight and I got five digits in there that's fine is that but I needed six one five six eight yeah I just went a little further than you needed and then it's going to multiply B so I just put the B with it and then 12 times one if I cross it up the other direction okay twelve times one gives me 12 and now I have to divide both sides by that nasty decimal zero point six one five six eight because that's going to cancel that out that's zero point six one five six eight is multiplying the B so to get rid of it we divide it zero point six one five six eight now I'll be able to figure out what B is as soon as I type that into my calculator so 12 divided by zero point six one five six eight and that tells me that it's nineteen point four nine nineteen point four nine months and so we found both sides using trigonometry now once I found one side I could have used Pythagorean theorem to find the other but we're practicing trig so I didn't example number two says find the missing side length now this is the last example I believe yep and then you got some homework questions from the textbook so let's see how quickly I can go in it says find the missing side length let's just do one side length this time let's find this one over here let's sign find sign little B so if we're gonna find side little B again we start by circling the angle that we're gonna use and then we're gonna label the sides well this points the way to the hypotenuse so that's the hypotenuse across from the angle I circled is opposite so it's the O and then the remaining side is called adjacent because it's right next to it's touching the 30 now if I'm finding a little B I'm going to switch colors here if I'm finding a little B I have to use little B and I have to use a side with a number on it so this one over here I don't need this at all there's no number on it and I'm not being asked for it so we're going to use side oh and side H now remember this so kah TOA well oh and H are with sign so we're gonna go sign and 30 degrees has to go with sign because that's we're gonna punch into the calculator equals opposite over hypotenuse so our opposite is B and our hypotenuse is 24 so sine of 30 and punch that into the calculator 30 and hit sign you might have to do it the opposite directional look at that that's a nice number 0.5 so sine of 30 is 0.5 equals B over 24 now this one is actually easier we don't have to cross multiply because the B's on top we can use isolation and just say oh I'm just gonna multiply by 24 and that gets me my be by myself but then I have to multiply by 24 on this side too and so my answer this side gives me the B and this side 24 times 0.5 well that's like saying what's 1/2 at 24 and the answer there is 12 and that actually concludes this video give it a try yourself