06 - Review of Essential Trigonometry (Sin, Cos, Tangent - Trig Identities & Functions)

hello welcome back to the physics 1 course this title of this lesson is called a review of essential trigonometry so every single physics problem that you're gonna work is going to seem challenging to you in the sense that no longer will the problem tell you exactly what you need to do to find the answer so a lot of times almost all of the time when we get past the basic problems there will be an angle of some kind in the problem maybe it's a plane like a wedge where a block is sliding down and the plane has an angle to it maybe a baseball is thrown at an angle maybe you're pushing with a force at 35 degrees so or whatever I can go on and on and on there's almost always gonna be an angle in every problem but you're never going to be told hey you need to do the sine of a cosine of the tangent of this angle you're never gonna be told that you're going to need to draw a picture and figure out from your previous knowledge of trig in order what to do you're gonna have to do that yourself and that's why these problems seem difficult so in this lesson we're going to review basic essential trigonometry it's important for you to know that I expect that you have seen this stuff before if you haven't then you need to stop and go to my trig class my trigonometry course and review the basic idea of angles and stuff and I am gonna review as we go through as we solve the problems I'm gonna baby step you as much as I can to review you but still you you need to have been exposed to this material so the first thing we're gonna do the most important things these are Lisa this is not a review of everything in trigonometry this is the review of the most important things to solve physics problems right so everything in this lesson is critical fortunately none of it is really hard so if you have a triangle remember this guy he talked about triangles all the time in trigonometry that's what it's all about so you have some kind of triangle and this is a right angle which means it's 90 degrees over in the corner and there's some angle like I told you you might have a wedge a lot of times in physics problems we have a wedge and we call this angle theta so theta don't let it scare you it's just another variable like X or Y this could be 15 degree angle or this could be a 35 degree angle or a 70 degree angle or whatever of course is the angle increases the steepness of this wedge you know gets taller or whatever but this is a general diagram here now we label the different of this triangle we say that relative to this angle the opposite side of this guy we're gonna label it opposite OPP for opposite and relative to this angle this side right here is adjacent adj we call it adjacent so adjacent means the side kind of close to the angle opposite means the side opposite the angle and then the very longest side of every right triangle is called the hypotenuse and I know that you all know that from basic from basic math now also in basic math you learn something call the Pythagorean theorem don't forget Pythagorean theorem don't forget that with right triangles like this you always have the Pythagorean theorem at your disposal the physics problem will never say use the Pythagorean theorem to solve it it'll never tell you that you just have to know that you can use this for every right triangle Pythagorean theorem if the way you learn in geometry is C squared is equal to a squared plus B squared in this equation C is always the hypotenuse it's always the longest side of this triangle the other two since they're added together it really doesn't matter the jacent or opposite doesn't matter what you label them so the way you read this in terms of this triangle that we have right here since C is the hypotenuse will write it as hypotenuse squared is equal to and again I told you it really didn't matter so we'll just call it adjacent squared plus opposite what does this equation mean that means for any right triangle no matter what the angle here is whether it's 15 degrees 30 degrees seven degrees really slender triangle whatever it doesn't matter if you know the length of the hypotenuse and you square it and you know the length of this and this and you square them separately and then you add these together they will equal each other always so it allows you to find the third side of the triangle okay the reason I'm bringing this up is because we're gonna use it a little bit later but it's important for you to know that it applies to all triangles oh not all triangles all right triangles with a 90 degree angle like this one here all right the next thing we're gonna do we're gonna talk about the fundamental trig functions now in truth there are more trig functions than the ones we're going to list here but these are the trig functions that are most used in physics we come across anything more complicated than this I'll explain it as we go the first one I know that you all heard of it's the sine of some angle theta now in a real problem you'll put a number here like 45 degrees or something but we're drawing everything generally so we say the sine of whatever this angle here is is defined to be the opposite side over divided by the hypotenuse so you literally take however long the opposite side of the triangle is and you divide by the hypotenuse all right that's defined as the sine of a single notice that the sine of the angle involves the opposite side okay remember that because that's going to be important the cosine of the angle of this triangle is equal to the adjacent side divided by the hypotenuse again notice that the cosine of the angle involves the adjacent side of the triangle that's going to be important so remember that so just but without going any further the sine of this angle involves something to do with the opposite side of a triangle the cosine of the angle has something to do with the adjacent side of the triangle they both involve the hypotenuse but that's kind of not so important for right now The Ting well it's important but it's not gonna be not trying not trying to make you remember that right now the tangent of the angle which actually we're not going to use quite as much in physics problems most of the time you'll probably use one of these you'll see why as we solve vector problems is defined as the opposite side divided by the adjacent side so it's the opposite divided by the adjacent side so these are critically important if there's one thing that I would write down in this review of essential trig first what's called a review I expect you have seen this before and it's essential trig which means it's not everything it's the most important stuff for these kinds of problems is I would want you to know these three things sine cosine tangent if that's all you remember that sine is opposite over hypotenuse cosine is adjacent over hypotenuse tangent is opposite over adjacent if that's all you remember then you can solve great many problems because most everything else that I'm going to talk about in this lesson is just going to be messing around with these functions to do useful things so if you only have to remember one thing remember these and understand what it means I know a lot of you looking at this sakes sine-theta what does that really mean it's literally just a fraction sign of whatever this angle is don't even worry about what the label is it just means that some thing that's related to this angle is defined as a fraction undefined on this triangle and if you forget which is which opposite adjacent whatever then just remember remember how to put this picture together and draw this picture on your test for many years I had to draw right triangles and label everything so that I would keep it straight now I just remember it but if you need to draw that picture please do draw it now we're gonna take this and we're gonna do something practical with it on the other board here so let's solve a quick little problem this is not exhaustive but we're just gonna do a quick little problem just to illustrate what I'm talking about here's a triangle like this now inside this triangle I'm going to again I do not know what this angle is sometimes I do know what the angle is but in this case I don't but what I do know now is that this side is three meters long this guy is four meters long and this guy is five meters long and I'm saying this is a right triangle first of all you can always verify this as a right triangle you should always make sure this is a right triangle how because we know that C squared is equal to a squared plus B squared this is true for any triangle that has a 90-degree angle in it C school C is the longest side so that's five now we have to square it and we're gonna check and see if it's equal to this squared plus this weird notice that a and B doesn't really matter because C I did four squared plus three squared if I called a equal to three instead then it would still be 3 squared plus four squared so the right hand side would be the same no matter what so the only one you really care about is to make sure the longest side goes into C's column here so this is 25 and this is 16 and this is 9 so 25 equals 25 check if you ever get an inequality like if you have 25 and this we're got 22 then something's wrong because the triangle has to be a right triangle in order to even use these sine cosine and tangent definitions and if this thing doesn't work out then this is not a right triangle because the Pythagorean theorem always is true for a right triangle okay so this was just an aside I wanted to show you a Pythagorean theorem what I mostly want to do is I want to ask you the question what is the sine of theta so literally when this happens you go back to your definitions which we just wrote now so I'm going to write it down again the sine of theta is equal to you remember I want you to remember if there's one thing in this lesson remember that the sine of an angle involves the opposite side the cosine of angle involves the adjacent side you're gonna find out why that's so important as we solve problems the sine involves the opposite side of this triangle and of course this divided by the hypotenuse okay opposite side opposite to what its opposite of this angle so that means it's three the hypotenuse is the longest side which is five all right so what this means is that the sine of this angle is equal to three fifths of an exact fraction if you put that in your calculator you'll find the zero point six so if I asked you what is the sine of theta you would circle this the sine of theta is equal to 0.6 alright let's go work the next part of this little simple problem and then I'll circle back and draw some conclusions at the end so that was for sine of the angle let's calculate for Part B let's calculate the cosine you know let's do it like this let's do like this I want you to find the cosine of the angle that's the question there so what is the cosine of the angle and I told you before the cosine involves the adjacent side so sine of all opposite cosine involves adjacent try to burn that in your mind so you're gonna write that down adjacent and it's gonna be divided by the same thing the hypotenuse so you go back to your triangle the adjacent sides of this angle is 4 and you're dividing by the same hypotenuse 5 so it's an exact fraction 4/5 so you would write down that the cosine of the angle theta is exactly 4/5 and if you want to put that in your calculator you can 0.8 that's the answer so notice first of all before we go any farther the sine of the angle came out to be some kind of decimal less than 1 and the cosine of the angle also turned out to be some decimal less than one that's always going to be true right so the sine when you if you go take your calculator out right now make sure it's in degrees degree mode because they're different modes in the calculator we're gonna be using degrees in physics almost all the time if you take any degree angle you want stick it in your calculator and hit the sine button you're always gonna get a number less than one if you stick any degree number you want any number you want and put it in the calculator and hit the cosine button you're always gonna get a number less than one that's gonna be important to understand when I wrap this all up in a nice and Bo at the end but you should always get decimals for these guys why because they're fractions and because we know that the hypotenuse is always the longest side of this triangle and the hypotenuse is always gonna be on the bottom then whatever on the top is always gonna be smaller than the bottom and that means that the fraction will always be less than one so sine and cosine ever get bigger than one ever that's the bottom line all right what is let's look at Part C what is tangent of this angle tangent is defined as a mixture of the two tangent is the one that involves the opposite side but it also involves the adjacent side but there's no hypotenuse anywhere here so we just write down the exact thing that we we know it's the opposite whoops I'm almost type wrote down in Jason it's the opposite side over the hypotenuse opposite side is three Hutt's not over hypotenuse opposite over adjacent sorry about that opposite over adjacent opposite is three adjacent is four so three fourths and you all know that that's 0.75 so you would write down the two tangent of some angle theta is 0.75 or you could write it as three fourths that's fine too when they're exact decimals like this I'm okay with decimals so you can go put it there all right so now we know what the sign of the angle is what the cosine of the angle is what the tangent of the angle is also notice the tangent of the angle is also less than one okay for this example the tangents are less than one now it turns out that the tangent function is is not always in between is not always like less than one or whatever the tangent function can go off the rails when you plot it I'll explain that some other time but for sine and cosine they always stay between plus one and minus one okay I didn't graph it for you but it's plus one and minus one this guy can kind of go all over the place and I don't want to get into the reason why right now but that's just something you can observe when you plot it so let's find out the most important thing here what is the angle theta in other words is this angle 13 degrees is this angle 17 degrees is this angle 35 degrees what angle is it the first thing I want you to know before we actually calculate the angle is this angle is locked in place by the distances of this triangle in other words if you got some string out or a pencil and you measured five and you measure four and then you went up vertically and measured three there would only be one way that you could put those together to make an actual triangle that connected that one triangle would have an angle here that would be fixed it's fixed because the size of the triangle are locked in place and you arranging them in one way because they have to make a right triangle so for any given set of sides there's only one angle there's only one answer I just don't know is it 35 degrees or is it 42 degrees what is it and so the way I figure it out is there's lots of different ways but the first one is let's go back up here we calculated what the sine of the angle and the sine of the angle here was and it was 0.6 all right so we know that the sine of this angle is 0.6 we calculated that before so remember from equations from equations like algebra equations you do the opposite to solve for X right you might divide by something multiplied by something add by something to get X by itself but the angle that we want to find is wrapped up in a sine how do we do that well we say well know theta is going to be equal to we have to do the inverse function in other words we have to undo it with an opposite so if sine is the function inverse side that's what the negative 1 means this is inverse sine or arc sine you might's undoes it so like the opposite of addition is subtraction the opposite of multiplication is division when solving equations the opposite of squaring something is a square root the opposite of sign is inverse sine right so when we do inverse sine to both sides of this thing in the first sign on the Left undoes it so we have theta inverse sine on the right so what we're writing here so if you stick to point six in your calculator and hit the inverse sine button you're gonna get the angle back and the angle in this case is going to be 36 point 87 degrees so that's the angle so if you built this triangle this would be thirty six point eight seven degrees and it can't be any other angle or else you can't make the triangle to begin with now we use the calculated theta using the sine okay but we can actually calculate it using the cosine because we also learned for this triangle that the cosine of the angle is zero point eight so if we want to get theta by itself we have to inverse cosine both sides that eliminates that kind of annihilates the left-hand side leaving theta by itself it'll be the inverse cosine zero point eight so if you put zero point eight in your calculator and find the inverse cosine button and hit that one what do you think you're gonna get you're gonna get thirty six point eight seven degrees thirty six point eight seven degrees notice it matches exactly everything self consistent whether you use the sine to find the angle or the cosine to find the angle you get the same thing now what do you think's going to happen if we use the tangent so let's go here just kind of squeeze it in the bottom if tangent of the angle is 0.75 then the the angle should also be what do you think the inverse tangent of 0.75 so if you put point seven five then find the inverse tangent button what do you think you're gonna get you're gonna get thirty six 0.87 the purpose of this example was two things to show you how to calculate sine cosine tangent and to show you that you can take any one of these things and do its inverse to find the angle all right because you're gonna be finding the angle a lot a lot of times it'll be here's what a baseball is doing what is the angle what angle did you throw it at well u ventually you're going to construct a triangle and you're going to end up having to figure out what this angle is and you'll use one of these so the second part of this is to tell you that it doesn't which one you use they all give you the exact same answer so there's not so much a right way to find the angle it's just there's about 16 different ways not literally but there's a bunch of different ways to find the angle okay the only other caution I want to throw at you when you're dealing with triangle trig this is this angle is it when you put these numbers in the calculator like point eight and find the inverse cosine you do this inverse tangent and so on when you do this stuff the angle that your calculator is gonna give you is always going to be the positive angle it's always going to be the angle as if the triangle was drawn here and it's getting give you this positive and this is a positive angle because it's measured from the x-axis like this this is x-axis y-axis so this is a positive angle right as we go and do more problems I'm going to caution you when your inverse tangent thing or inverse cosine or inverse sine you got to be a little careful because you have to know that that calculator is always going to give you the positive angle because sometimes what if I'm throwing the ball down right so the triangles not oriented up like this it's oriented down then that means that I'll actually have a negative value for the Y and a positive value for the X and so long story short you've got to be careful to look at the quadrants of what you're actually doing when it returns that angle you just have to know that it's always going to give you that positive angle but your problem might actually be throwing the ball the other way and you might actually have to add 180 degrees to that angle or something like that to get the correct angle you want but well when we get to that point I will explain and caution you as we do more problems like that for now just know when you inverse cosine inverse tangent you're always going to get that positive angle back that's all I want you to remember at this point all right so this is very important coming up next if you have a triangle as we have been talking about and it has some angle theta and it has an opposite side and it has an adjacent side and it has a hypotenuse as they all do I'm kind of regurgitating over and over again that we had talked about the fact that the sine of the angle was equal to the opposite remember sine deals with opposite for hypotenuse and we talked about the fact that the cosine of the angle had to do with the adjacent side remember cosine deals with adjacent over hypotenuse absolutely true but there's actually a little bit I want to call it easier but it's a very very useful way to write this down if you were to take this and solve it for the opposite side how would you do that well this is a fraction you have to multiply left and right by the hypotenuse right so the if you solve this equation you're gonna see that the opposite side is equal to the hypotenuse times sine theta okay let me do this one and then I'll explain what I'm talking about here also with this one the adjacent side is equal to the hypotenuse times the cosine of theta these equations are the same thing as these there's no different I'm not introducing anything new I'm just telling you that if you take this relation and you solve it for the opposite side you'll have to multiply both sides by the hypotenuse so it cancels on the right giving you the opposite side here you multiply by the hypotenuse same thing here here you multiply by the hypotenuse alright these are very useful they are the exact same thing as these of course they are but they are very useful and the reason they're useful is because in physics we deal with something called a vector I'm actually gonna explain really briefly what a vector is to you now but then we're gonna have an entire lesson actually two or three lessons on vector so don't stress out if you don't get everything right now but the breathe the point is these particular relations are very very useful let me explain why they're useful because this is one of the very first things you'll be doing in physics let's say I throw a ball this direction right how fast do I throw it 10 meters per second obviously I'm not throwing it horizontally I'm throwing it up with some angle so what I do is I define the angle and the way I define an angle as I draw a right triangle and I'm always different throwing the ball relative to the ground this is the ground here so let's say I throw that ball at 35 degrees let's put numbers in here instead of just theta okay you all know that if I throw a ball at an angle up I have to throw it kind of in two directions right I'm flirt when I do that I'm throwing it horizontally that's the horizontal speed and I'm also throwing it vertically straight up and down the mixture of those two motions horizontal and vertical is what gives you the I know I know the path curves forget about the curving but right when it leaves your hand it's going at an angle which is a mixture of those two and that mixture is reflected in the triangle you can see that if I should throw it at 35 degrees that the horizontal component of the velocity is bigger than the vertical component of the velocity so I might have a vertical speed right I know I've been throwing around the words for speed and velocity will define these terms a little bit but I think all of you know that what's being speed more or less is or not so speed okay so the reason I'm showing you and telling you that these relationships are very useful is because of the following thing usually I throw a baseball this arrow represents how fast I'm throwing that ball in total in the angle direction 10 meters per second that's what that means 10 meters every second that ball goes that's pretty fast that's faster than I can throw but anyway it's a nice number okay but let's say that I want to figure out what is the vertical part of the speed in other words how much of the speed exist only in the up and down direction and how much of the speed exists only in the horizontal direction because the angled speed is a mixture of the two all right so here's what you do the vertical speed notice in this triangle the vertical speed is the opposite side of this triangle to the angle notice that this equation tells me the opposite side of this angle or the opposite side of the triangle is the hypotenuse times the side right so I'm trying to find the opposite side of this triangle which I know from this equation is the hypeeeee News times the Sun okay which means that the opposite side of this triangle is equal to what ten times the sine of 35 degrees right so the vertical speed is equal to if you take sine 35 and hit that in your calculator and you multiply the answer by 10 you get five zero point seven four meters per second it's the same units as whatever I threw it with let me kind of leave that alone let's calculate the horizontal speed and then I will wrap it all up together and impress upon you something that you really must remember okay so the horizontal speed horizontal speed what is that that's the according in this triangle that's the adjacent side of this triangle that's what I'm trying to find the adjacent side of this triangle but we just said from this equation from the definition of cosine I can find the adjacent sine of any triangle hypotenuse times cosine so it's the hypotenuse times the cosine of now I know the angle is 35 degrees right and so let me switch colors to kind of put the point home so the horizontal speed is equal to what it's well the hypotenuse here is 10 so let me go ahead and just write it down here so it's 10 times the cosine of 35 degrees so if you take 35 degrees hit the cosine button multiply by 10 you'll get eight point one nine meters per second now first of all let's see if this makes sense we're saying that we're throwing a baseball at ten meters per second at some angle and since the angle is pretty small so less than 45 degrees we're saying just from the triangle the way it's drawn the horizontal speed should be larger than the vertical part of the speed the horizontal speed should be larger than the vertical part of the speed and that's true so why did I bring this up because the probably the most important thing in trigonometry that you will learn in physics and this is why I'm turning around and I'm looking right at you because I really really want you to remember this is we use triangle trigonometry to take what we call vectors and break them up into what we call components this ten meters per second is called a vector please don't let the word scare you vector is not a complicated thing it means I throw a ball at a certain speed I mean you know that that's not that hard to know the length of this arrow is 10 okay but maybe I don't want to just know what that angled length is I want to know how much is in this direction horizontal and how much is in this direction vertical so we take that vector and we break it into components meaning a horizontal direction and a vertical direction and we use sine cosine tangent every single time to do that mostly sine and cosine to do that okay so here's the equations that way I taught you and then we applied down here the opposite side of a triangle is hypotenuse times sine the adjacent side of the triangle here is hypotenuse times cosine hypotenuse is the total speed in this example 10 meters per second that I've thrown the baseball here's what I have written in my notes I'm not going to write it down I'm just gonna say it about four times to make sure you understand when we use a sine function and multiply it by the total length of the baseball in the speed in this case what the sine is doing listen to me here is it's taking the dolt total speed ten meters per second and it's chopping it is what I have actually this is the way I think about it it's chopping it down you throw the ball at ten meters per second but I want to chop it and I want to know only how much is going in the vertical direction that's the opposite side going up so when you take a sine of an angle and multiply it by the hypotenuse what you're doing is you're taking the total speed and you're chopping it finding how much of it goes in the vertical direction because vertical is opposite remember I told you sine always deals with opposites and cosines always deal with adjacent and that's to help you remember that when you take the sine of this angle or multiply it by this it takes this number and it chops it only to give you the vertical part of the speed right and when you take this number and take its cosine that deals with with adjacent so when you take the cosine of this and multiply it by the same thing you're chopping it also but only giving the horizontal part which is this so here's the moral of the story because in this case I've given you a speed at 10 meters per second at an angle but this vector thing but we haven't talked much about it can represent lots of things it can represent a force maybe I'm pushing on something at an angle and I'm pushing with a hundred pounds of force or 100 Newton's of force we'll talk about Newton's later it's the unit of force okay but I want to know not how much am i pushing at an angle how much am i pushing horizontally and how much am i pushing vertically well I do that by using triangle trig I take this when I multiply it by the cosine of this it chops it and gives me only the horizontal part when I take this and I multiply it by the sine of this it chops it and it only gives me the vertical it's crucially important because all these physics problems what we're gonna do we're gonna break them all apart into X direction and we're gonna solve those and then we're gonna break them separately into Y direction and we're gonna solve those we're gonna break everything apart and solve the different directions separately it's important to be able to take this and chop it this way and chop it this way to get the two different components so here is your mega mega mega summary of everything we learned in this section summary summary all right here's the summary when we have a triangle like that the sine of an angle is the opposite whoops the opposite over the hypotenuse the cosine of an angle is the adjacent over the hypotenuse the tangent of an angle is the opposite over the adjacent so it doesn't involve a hypotenuse there okay now we can take these guys and we can we can find use these equations to find the angles in every one of these cases how do we do it we can say that the angle is equal to the inverse sine of the exact same thing opposite over hypotenuse we can say that the angle is also equal to the inverse cosine of the adjacent over the hypotenuse and we did an example showing exactly that and we can say that this angle is also equal to the inverse tangent of opposite over adjacent literally you take the fraction stick it in there if you want to find the angle you just inverse it that's all you do okay which one of these is correct they're all correct you can find the angle using any one of these relations and we actually showed you that a minute ago and then the final thing I want to point out is that we can take these equations so that's a useful set of relations we can take these set of equations and we can write even more useful set of relations we can say that the opposite side of the triangle is just the hypotenuse times the sine of the angle that's going to give you the vertical if you kind of think of the triangle the way we've been drawing it and the adjacent side of that triangle is the hypotenuse times the cosine of the analyse the sign chops this thing to give you the vertical part and the cosine chops this angle if you're multiplying by the hypotenuse giving you the horizontal part so when you look at a trig book or a physics book you'll see these equations you'll see these equations and you'll see these equations and it gets very confusing which ones do I remember really all you really need to remember are these these are the core equations these just come from these these just come from these so really just remember these but what's gonna happen is you're gonna use these so much that you will remember them to find the vertical part of a vector you're gonna take the hypotenuse and chop it with a sine chopping with a sine gives you that vertical part to find the horizontal part of some vector some arrows and hypotenuse of a triangle you're gonna take the hypotenuse and you're gonna chop it with the cosine that gives you the adjacent or the horizontal part those are the core things of trig when you take a trade course you'll do so much more than what we've done here you will plot sine cosine and tangent as and you'll see that sine and cosine go up and down like waves and you'll see that tangent looks even crazier and there are other trig functions besides sine cosine and tangent that you learn in a trig course in a trade course you'll also learn or a precalculus course you'll learn about the unit circle and you'll learn about how to calculate sine cosine and tangent using the unit circle and you'll learn about radians and you'll learn how to convert Radian measure to degrees because that's a unit of measure of angle also but in physics which is when I'm focusing on the the most important things these things are important what if you don't understand this watch it a few times if you don't understand anything about it then go review trig using my trig and precalculus class and then come back here and continue learning physics with me we'll do everything step by step and give you plenty of practice to do really well and get a deep understanding of the topics

Channel: Math and Science

Views: 60,191

Rating: 4.9202952 out of 5

Keywords: trig, trigonometry, trig review, intro to trig, trigonometry introduction, trigonometry intro, trigonometric functions, trigonometry basics, trigonometric equations, trigonometric ratios, pythagorean theorem, sin, sine, cos, cosine, tan, tangent, trig sin cos tan, trig sin cos tan identities, trig sine rule, trig sine function, trig cosine rule, trig cos sin, trig cos rule, trigonometry cosine rule, trig functions, trig identities, unit circle, trig formulas, trigonometry formulas

Id: OVVh6enaQcs

Length: 33min 33sec (2013 seconds)

Published: Tue Aug 13 2019