Maths Tutorial: Trigonometry SOH CAH TOA (trigonometric ratios)
The trigonometric ratios is talking about the sine ratio the cosine ratio and the tangent ratio and what it's talking about is when we have a right-angled triangle we can use sine cosine and tangent to figure out sometimes an angle within the triangle and sometimes a side of the triangle if we're given enough other pieces of information the trick is to label your sides correctly first off we must always have a right angle to be able to apply this if you have a triangle that's not a right angle like that or like that they kind of look the same if you have a non right angle triangle you can't apply this so that's the first thing you have to check for always opposite your right angle is the longest side along the triangle and the longest side which is opposite that right angle we call the hypotenuse the other two are going to be called the opposite and the adjacent and that'll depend in relationship to which angle we're talking about so say our mystery angle theta for example is down here the angle this angle in the triangle the side that's opposite it is this one if we spray outwards from that angle which line do we form we form this line here so that being opposite our angle is the opposite side the side that's left over not already labeled is the one that's next door to our angle and because it's next door it's next to it it's adjacent to it so we call that the adjacent side now if we were talking about a different angle say this one up here instead of this one then our sides would be labeled differently the hypotenuse would stay the same because the hypotenuse is always the hypotenuse it's always that one that's opposite the right angle but now if we're talking about this mystery angle up here the side that's opposite it that it creates by that angle that angle splays out and becomes this side down here so now this is the opposite and the one left over the one next door over here would be the adjacent so that's how we label our triangle now how do we apply these ratios the way these ratios work this top part of the mnemonic that we're using to remember this sohcahtoa stands for the sine of theta equals the opposite over the hypotenuse so in our s0h we have the sine equals the opposite over the hypotenuse for car we have the cosine equals the adjacent over the hypotenuse that's our C a H and for this part the Toa we have the tan equals the opposite over the adjacent that's our Toa so for the sine of theta we have the opposite over the hypotenuse for the cosine of theta we have the adjacent over the hypotenuse and for the tangent of theta we have the opposite over the adjacent so say you've been given the information that the angle down here is 43 degrees and this side which is the hypotenuse is 12 and we're trying to find the length of this side going along here well because it's a right angle triangle we can use sohcahtoa to work this out now this X is that the opposite or the adjacent we look at our angle the piece of information that we have and we figure out which line is that angle creating its splaying out towards something and it's creating this line here so it's opposite that angle which means this is the opposite this side over here is the hypotenuse which means that this one leftover is the adjacent so we don't have any information to do with the adjacent so we're not applying that we're going to be using the opposite and the hypotenuse so we've got O H so which one of these three are we going to use we're going to use this one so we have the sign of our mystery angle 43 is equal to the opposite over the hypotenuse which is our x over 12 now to get X by itself I'm going to times this 12 over to the other side or you could call it x in both sides by 12 and then this cancels out either way we get sine 43 times 12 equals x so you just whack that into your calculator and you need to make sure your calculator is in degrees mode for this to work out if your calculator is in radians you'll get a different answer so if you don't know how to set your calculator to put it in the correct mode make sure you ask your teacher for this one we get an answer of x equals eight point one eight let's say now that we're given this angle up here is 27 degrees and we're trying to find this unknown over here so what we've got is the hypotenuse over here the side that is opposite our angle the side that is created by that angle is over here so that's the opposite down there which means that the piece of information we do have is the adjacent so we're going to be using the adjacent and the hypotenuse which we means we've got a H so which one of these are we using we're going to be using that the cosine one so the cosine ratio is that cos of theta equals the adjacent over the hypotenuse which means the COS of 27 is equal to five over Y now I need to get Y by itself the COS 27 and the Y are actually just going to swap places but I'll show you how that works first of all I need to move the wire away from the five so cause 27 times y equals five now to get the Y by itself it's been x by the cause 27 so I need to divide by that so I have y equals five divided by cos 27 and we get an answer of y equals five point six one say we're given an angle down here of 38 and we're trying to find this X the hypotenuse is over here and that doesn't factor in because that's not one of the two pieces of information that we do have so what we have is the side that's opposite our angle the side that our angle is forming by splaying out towards it that's part of it the opposite and the other one we've got is the adjacent so we're dealing with the opposite and the adjacent we've got o and a so which one of these are we using funnily enough we're going to use 10 so we have tan theta equals the opposite over the adjacent which in this case would be tan of 38 equals the opposite x over the adjacent 10 so to get the X by itself we times the 10 over here we have tan 38 times 10 equals x so x equals 7.
8 we can use this if we're trying to find an angle as well in this case we have an angle down here our mystery angle and we've got two of the sides so let's label them this one over here is the hypotenuse because it's opposite that right angle this angle splays out towards this line going down here so this three must be the opposite and I don't need to worry about the adjacent because it doesn't factor in I don't have that piece of information I've got the two I'm going to use so I've got the opposite and the hypotenuse which means that I'll be dealing with sine so sine of theta is equal to the opposite over the hypotenuse which means sine of theta is equal to 3 over 5 now how do I work out theta if I've got sine of theta here well what we do is we theta equals the inverse sine of 3 over 5 on your calculator this looks like a little sign to the negative one so it might be the second function above your sign button you'll probably press shift sign to get this little figure sign to the negative one that's taking the inverse sine basically so you do this with three point five in the brackets in your calculator and you get theta equals 37 degrees so we've got our mystery angle up here this over here let's do our labeling this would be the hypotenuse but that's not one of the two pieces of information we do have so we don't have to worry about that this angle splays out towards this line which means that this down here is the opposite and this is the one left over it's the next-door neighbor so it's the adjacent so we're dealing with the opposite and the adjacent we've got oh and a so we're going to be looking at 10 so tan theta equals the opposite over the adjacent which means tan theta is equal to 155 divided by 70 so to find theta we take the inverse tan of 155 over 70 which gives us theta equals 66 degrees so in this triangle what is the length of PQ and remember when they say two letters like that they're talking about the line that joins those two points so we're talking about this line along here this is line from P to Q so this is our X down here so the first thing we need to check is is it a right angle triangle because if it's not a right angle triangle then we can't use sohcahtoa but what do you know funnily enough the example I use in the sohcahtoa video is a right angle triangle so this is our hypotenuse over here because this is opposite our right angle what about the other ones here is our mystery angle and the line that it's splaying out to create is this one over here so this would be our opposite which means our left over down here is the adjacent the piece that I'm trying to find is the adjacent it's this X and the piece of information that I've been given is the hypotenuse so I'm dealing with a and H so in sauk cut Toa which one gives me an A and an H there it is eh so I'm using the cosine ratio so I'm using the cos of theta mystery angle equals the adjacent over the hypotenuse and that's where I get my see a H from in the car so cos of the angle that I have is 42 equals the adjacent side which is my x over the hypotenuse which is 14.
5 so to get the X by itself on one side of the equals so I can solve X I'm going to get rid of this 14.5 it's been divided so i'll times it over here I have cause a 42 times 14.5 will give me X whack it into your calculator I get x equals ten point seven eight and it's important to put the units on there that's going to be centimeters what about this one I've got a right angle triangle so I know I'm going to be able to use these trigonometric ratios now which one's the hypotenuse it's the one that's opposite the right angle so that's going to be over here so there's my hypotenuse there for the six point two the angle that we're dealing with is over here and the line that it creates by splaying out to make a line the one opposite is over here so this is our opposite and this must be the next-door neighbor this is the adjacent so I've got the a and the H again as the pieces of information so I'm going to be using cos again cos of theta equals a over H but this time I'm not solving for a side I'm solving for an angle itself so I say cos of theta equals the adjacent 3.
8 over the hypotenuse six point two and to find theta I take the inverse cause that's like undoing this cause operation to get it over that side what's happened to the theta well we've kind of caused it so we're gonna uncover it by taking the inverse cos of three point eight over six point two I think I've gone off the side of the screen there sorry work that into your calculator and you get 52 point two degrees last example a flagpole is secured by guy ropes and kid to the ground eight meters from the base of the flagpole and to a point nine meters up the flagpole find the angle the guy ropes make with the ground and the length of the guy ropes okay so the flagpole secured by curbs anchored to the ground eight meters from the base of the flagpole so these lengths here are eight meters from the base of the flagpole in the center so from here to here is eight and from here to here is a the flagpole itself or where these ropes join up at least that length there whoa not a straight line is nine meters and do we have a right angle or not even though it looks like maybe this triangle isn't a right angle we can actually assume that we do have one because what we're measuring is off this line right here in the center at the center of that flagpole and that makes a right angle going this way so here's my triangle and I've got a right angle and I know two of the sides so first of all we need to find the angle the guy rope makes with the ground so that's this angle there so we're trying to find the angle between the rope and the ground which means that's our angle so hypotenuse comes out from the right angle that's that one there the opposite comes out from our mystery angle so that's that one there and the one that's left over is the adjacent so that's that one down there meaning the.
Two pieces of information that I have the nine and the eight are the a and the O so I've got o a I'm going to use 10 so 10 of my mystery angle is equal to the opposite which is 9 over the adjacent which is 8 so theta equals the inverse tan of 9 over 8 so theta equals 48 point four degrees now for Part B I'm going to find the length of the ropes so now what I'm trying to find is that length there which is the hypotenuse so now I'm trying to find the hypotenuse now I need to be either finding an angle or at least know one angle within the triangle I can't use o-a and haitch because then no angle factors in so what I'm going to do is that the fact that I've got this piece of information down here I'll use that as one of the bits of info that I know and I can either use the O or the a that is the nine or the eight to form the ratio so what I mean by that is I could say the sine of 48 point two four is equal to the opposite which would be 9 over the hypotenuse which is my mystery angle and work it up that way or using the a instead of the o I would say the coz of forty eight point four is equal to the a which is eight over the hypotenuse put either of those into your calculator and you'll get an answer of H equals 12 meters.
8 we can use this if we're trying to find an angle as well in this case we have an angle down here our mystery angle and we've got two of the sides so let's label them this one over here is the hypotenuse because it's opposite that right angle this angle splays out towards this line going down here so this three must be the opposite and I don't need to worry about the adjacent because it doesn't factor in I don't have that piece of information I've got the two I'm going to use so I've got the opposite and the hypotenuse which means that I'll be dealing with sine so sine of theta is equal to the opposite over the hypotenuse which means sine of theta is equal to 3 over 5 now how do I work out theta if I've got sine of theta here well what we do is we theta equals the inverse sine of 3 over 5 on your calculator this looks like a little sign to the negative one so it might be the second function above your sign button you'll probably press shift sign to get this little figure sign to the negative one that's taking the inverse sine basically so you do this with three point five in the brackets in your calculator and you get theta equals 37 degrees so we've got our mystery angle up here this over here let's do our labeling this would be the hypotenuse but that's not one of the two pieces of information we do have so we don't have to worry about that this angle splays out towards this line which means that this down here is the opposite and this is the one left over it's the next-door neighbor so it's the adjacent so we're dealing with the opposite and the adjacent we've got oh and a so we're going to be looking at 10 so tan theta equals the opposite over the adjacent which means tan theta is equal to 155 divided by 70 so to find theta we take the inverse tan of 155 over 70 which gives us theta equals 66 degrees so in this triangle what is the length of PQ and remember when they say two letters like that they're talking about the line that joins those two points so we're talking about this line along here this is line from P to Q so this is our X down here so the first thing we need to check is is it a right angle triangle because if it's not a right angle triangle then we can't use sohcahtoa but what do you know funnily enough the example I use in the sohcahtoa video is a right angle triangle so this is our hypotenuse over here because this is opposite our right angle what about the other ones here is our mystery angle and the line that it's splaying out to create is this one over here so this would be our opposite which means our left over down here is the adjacent the piece that I'm trying to find is the adjacent it's this X and the piece of information that I've been given is the hypotenuse so I'm dealing with a and H so in sauk cut Toa which one gives me an A and an H there it is eh so I'm using the cosine ratio so I'm using the cos of theta mystery angle equals the adjacent over the hypotenuse and that's where I get my see a H from in the car so cos of the angle that I have is 42 equals the adjacent side which is my x over the hypotenuse which is 14.
5 so to get the X by itself on one side of the equals so I can solve X I'm going to get rid of this 14.5 it's been divided so i'll times it over here I have cause a 42 times 14.5 will give me X whack it into your calculator I get x equals ten point seven eight and it's important to put the units on there that's going to be centimeters what about this one I've got a right angle triangle so I know I'm going to be able to use these trigonometric ratios now which one's the hypotenuse it's the one that's opposite the right angle so that's going to be over here so there's my hypotenuse there for the six point two the angle that we're dealing with is over here and the line that it creates by splaying out to make a line the one opposite is over here so this is our opposite and this must be the next-door neighbor this is the adjacent so I've got the a and the H again as the pieces of information so I'm going to be using cos again cos of theta equals a over H but this time I'm not solving for a side I'm solving for an angle itself so I say cos of theta equals the adjacent 3.
8 over the hypotenuse six point two and to find theta I take the inverse cause that's like undoing this cause operation to get it over that side what's happened to the theta well we've kind of caused it so we're gonna uncover it by taking the inverse cos of three point eight over six point two I think I've gone off the side of the screen there sorry work that into your calculator and you get 52 point two degrees last example a flagpole is secured by guy ropes and kid to the ground eight meters from the base of the flagpole and to a point nine meters up the flagpole find the angle the guy ropes make with the ground and the length of the guy ropes okay so the flagpole secured by curbs anchored to the ground eight meters from the base of the flagpole so these lengths here are eight meters from the base of the flagpole in the center so from here to here is eight and from here to here is a the flagpole itself or where these ropes join up at least that length there whoa not a straight line is nine meters and do we have a right angle or not even though it looks like maybe this triangle isn't a right angle we can actually assume that we do have one because what we're measuring is off this line right here in the center at the center of that flagpole and that makes a right angle going this way so here's my triangle and I've got a right angle and I know two of the sides so first of all we need to find the angle the guy rope makes with the ground so that's this angle there so we're trying to find the angle between the rope and the ground which means that's our angle so hypotenuse comes out from the right angle that's that one there the opposite comes out from our mystery angle so that's that one there and the one that's left over is the adjacent so that's that one down there meaning the.
Two pieces of information that I have the nine and the eight are the a and the O so I've got o a I'm going to use 10 so 10 of my mystery angle is equal to the opposite which is 9 over the adjacent which is 8 so theta equals the inverse tan of 9 over 8 so theta equals 48 point four degrees now for Part B I'm going to find the length of the ropes so now what I'm trying to find is that length there which is the hypotenuse so now I'm trying to find the hypotenuse now I need to be either finding an angle or at least know one angle within the triangle I can't use o-a and haitch because then no angle factors in so what I'm going to do is that the fact that I've got this piece of information down here I'll use that as one of the bits of info that I know and I can either use the O or the a that is the nine or the eight to form the ratio so what I mean by that is I could say the sine of 48 point two four is equal to the opposite which would be 9 over the hypotenuse which is my mystery angle and work it up that way or using the a instead of the o I would say the coz of forty eight point four is equal to the a which is eight over the hypotenuse put either of those into your calculator and you'll get an answer of H equals 12 meters.