Intro to heights and distances

to get introduced to heights and distances let's imagine that you are Superman and in front of you there's a building and on the top floor of that building there's somebody who needs your help an innocent citizen trapped by a cruel villain a generic Superman story and now if your normal man then maybe you'll run all the way here take an elevator and then go there but you're Superman so you decide to fly straight over there and you want to know how far you have to go to reach there in other words you want to find the length of this line and there are many ways you can find it but because you're Superman you come up with a way that you can do it really quickly and one of the features that you have as Superman is that I don't know if for the ones were not familiar with Superman so one can shoot lasers from his eyes whenever he wants to so now you can shoot lasers from your eyes now the thing about this laser is that it shoots exactly in the direction that you're pointing your eyes at that moment so if you're seeing straight ahead without tilting your head up or down then this laser points exactly along the horizontal now okay how can this be useful to find how far you should fly and that's where things get clever so what you do is that you start tilting your head up up up up up until you reach the top of the building your laser reaches the top of the building and it's pointing exactly to where you want to go and then what you do is that you find out how much you have to turn your head that's all you did you didn't have to go fly there all you did was find out how much you have to turn this laser from the horizontal position till it points towards the top of the building and you found that this angle is 45 degrees now that's a lucky coincidence because this could have been sixty three point to 75 degrees but in this beautiful world he got it to be 45 now the question to you is how will you use the length that you are from the building the distance that you are from the building that that Google Maps is telling him and this angle that you have to tilt your head such that how much should you have to tilt your head such that the laser points there using these two how can you find the length of this of this line segment now notice here that the thing to notice here is that this building makes a right angle with the horizontal that's the key idea over here which means that you have a right angles triangle over here now that you can see a triangle you can ask okay I know one angle I want the adjacent I actually want the hypotenuse I know the adjacent what ratio connects the adjacent and the hypotenuse and you know that that ratio is called something it's called cos so if you have cos of 45 if you remember or you can look up in a book the COS of 45 then you will be able to find this distance and which is what you want so let's write D over here and maybe our little book wave written in the course of 45 the way I remember it is that I remember what I call the 45 triangles I'm going to draw it over here 345 triangle by that I just mean a right triangle that has a 45 in it and I know that if this side of this is 1 this also has to be 1 that's what I actually remember because this is 45 and this is a triangle then this must be 45 which means it's an isosceles triangle these two sides have to be equal and then I can use Pythagoras theorem to just write root 2 over here through the 1 square plus 1 square and with this triangle in my mind I can remember or look up look up all the ratios that I need sign will be sine 45 will be 1 by root 2 cos 45 will be 1 by root 2 in the case of 45 these two are equal and tan will be 1 cot will also be 1 so I can do that and in this case what I need is cos 45 so what I do is that I write okay cause of 45 cos of 45 degrees must be equal to the adjacent by the hypotenuse no matter what the triangle what size the triangle is they're all gonna have the same ratios that's why these remembering these ratios are useful so hundred by D and now I can replace cost of 45 with 1 over root 2 1 over root 2 and now of course you can just find the answer if you cross multiply you will get D over here and hundred root 2 over here so D equals 100 root 2 oh and what's the unit 100 root 2 meters 100 root 2 meters now you can leave it here I mean that's what I'm tempted to do but if you want to find what they were the answer is row 2 is approximately 1 point 4 and 4 so 100 root will be 140 one point four meters 140 one point four meters so there it is you have it now why did we talk about Superman and lasers shooting from the eyes we did that because you don't have to be Superman to imagine a laser shooting from your eyes Superman the laser will actually shoot but if you're not you can imagine a laser shoot to be shooting from your eyes exactly in the direction at which you're pointing your eyes at that moment and that line is what we call is a name for this line that line is what we call the line of sight line of sight so if you're not tilting your head above or below your line of sight will be exactly along the horizontal and then as you tilt your head above and above your line a side goes there your line is that is basically that laser that shoots from your eyes in the direction that you're seeing and then you keep tilting it and maybe you reach another point now your line of sight is over here now you have to tilt your head by 45 degrees from the horizontal right this angle is called the angle evaluation of this point as seen from you or seen by you so I'm going to write this the angle of elevation now notice notice why do we have a name like this right I can now say I give you a big story Superman shoots lasers from his eyes he had to turn his head 45 degrees from the horizontal such that his laser points her head at the top and I was a long story a much shorter way of saying this is that the top of the building has an angle of elevation of 45 degrees as seen from or as seen by Superman that conveys the exact same thing you will know okay this means that the big version of that is this means that a Superman kept us or anybody thus it doesn't matter it doesn't matter that it's so on if anybody we're at the horizontal they would have had to tilt their head by 45 degrees such that the laser from there I the imaginary laser from there I would point to the top of this building so this new word is really useful it's a short form for a long story and a lot of the questions that we will play with all happen to be stories that we will break down at diagrams like this and find the right triangles now this is a new word the line of sight this is another new word the angle evaluation we have one more new word it's called the angle of depression now you may be able to guess what that word should mean because you can see what angle evaluation means think about it let's play with the scenario to understand what angle of depression is now in the second scenario let's say Superman is on the top of the building he has finished saving and now he's relaxing and then there is a citizen walking their dog and they're about to collide with a bicycle now Superman wants to save them but the only thing you know this time is that the angle of depression that they have the angle of depression of them the citizen from Superman's point of view is 60 degrees that's all you know but you know what angle evaluation is so can you guess what this must mean what does it mean to say the angle of depression of them is 60 degrees as seen from here think about that now let's let's visualize this let's take an imagined Superman just relaxing so he has no idea so his line of sight is just along the horizontal he's just looking into the sky and what happens is that now that he has to look at them he has to turn his head down so he's gonna turn turn turn turn turn turn turn and then point towards them he's obviously not pointing the actual laser he's pointing towards them and how much did he have to turn his head that's what we call the angle of depression so this distance this angle actually this angle that he had to turn his head is what we call the angle of depression this angle is given to be 60 degrees over here and that's the angle of depression that's the new other new word that you may want to know the angle of depression this is a short form for saying from the horizontal you will have to turn your head 60 degrees downwards to see whichever point that you're talking about so if this point has an angle of depression of 60 it means you have to turn your head 60 degrees from this point to see this so it's a very short way of representing so the word depression tries to talk about turning or tilting down and the word elevation the angle of elevation talks about having to tilt your head up and both of these are always measured from the horizontal and now how do you find how far you have to fly you have an angle you don't have anything else so can you find how far you have to fly with just this angle even if you're Superman you actually can't so what Superman does is or you do if you're still thinking of yourself as Superman is that you quickly google the height of the building it's a famous building and you find that luckily the height of the building is hundred meters so we're actually choosing convenient numbers so that we're not distracted by the numbers when we're trying to learn some new concepts see 100 meters over here and you have 60 degrees over here what can you do you will notice that this is horizontal the ground is horizontal this has 90 degrees we will come to that this ground is horizontal which means this is a transversal that cuts both these parallel lines which means that this angle and this angle have to be equal this angle here have to be equal 60 degrees now why is that interesting that's interesting because now you have a right angle triangle your 60 degrees over here you know the opposite side of this triangle and you want the hypotenuse what connects the opposite in the hypotenuse sine right so if you know sine 60 you're done the way I try to remember sine 60 is that I remember what I call the 60 degree triangle what I mean by that is the triangle which has one 90 this angle to be 60 and then if this is one I always take my base to be one then the only thing I remember is that the hypotenuse will be twice the base for cos in other words cos 60 is half with this I can find the third side using Pythagoras theorem 2 square minus 1 square is 3 so this will be root 3 so all the ratios of 60 that I want and bonus because 30 is like hiding over here I have all the ratios that I want for 30 degrees as well so with just these two triangles I can remember all of the ratios that I want sine cos and tan of 30 45 and 60 and this is what I use and now we can find what this length is and we just kind of call it something I'm gonna call it D so 100 by D will be equal to sine of 60 which is root 3 by 2 so let's write that so 100 by D 100 by D now you can find okay with all this you can find D and you know that you can find it for any triangle that has 60 in it and what is D gonna be from this you can just cross multiply so D goes over here it's 200 by root 3 D equals 200 by root 3 now I'm just gonna leave it at that you can you can always find the answer to this so the key takeaway of this video is that all the questions that we will see will have three new words that we will use one of them is line of sight line of sight which is just the laser that shoots from your eyes pointing in the direction that you are seeing so if you're not tilting your head up or down your line of sight is exactly along the horizontal and then as you tilt your head down your line of sight is now pointing towards them and then the amount you had to turn from the horizontal to look at a point with your laser is what we call the angle and depending on whether that angle is below the horizontal or above we have two names if it's above we call it angle of elevation let's hope I have spaced your angle of elevation and if it's below like over here we call it the angle of depression these are the only new words that you need to know and that's it once you have these words and once you remember these two triangles because a lot of us think I have to remember a lot of formulae maybe because there are many questions not really the only two triangles you need to remember are these and if you remember these you have the basic ratios sine cos and tan of 30 45 and 60 so these are the prerequisites I like to call them the things you need to know to be able to solve problems and all of the questions boil down to taking a story that's given to you and drawing a diagram and asking where are the right triangles here