Angles of Elevation and Depression

[Music] everyone I'm mr. a and today I want to talk with you about an angle of elevation versus an angle of depression this is a very popular topic that comes up when you're studying trigonometry or geometry and a lot of times those acute those appear together and when we talk about an angle of elevation versus an angle of depression an angle of elevation is when we are looking up at something so think angle of elevation is looking up angle of depression is looking down now up and down a relative terms there's no such thing as up in an absolute sense right if you imagine right now if you all pointed up you'd be pointing up from the ground but if someone on the opposite side of the world was pointing up they'd be pointing in exactly the opposite direction as you so up and down are always relative to something in this picture here we've got us let's say that's us on top of this lighthouse and then we have our friend down here on the ground waving to us if we were to look at our friend we would of course I think all agree that we'd be looking down at our friend but looking down from where an angle appellative depression is when we look down from horizontal so it's not just looking down it's looking down from horizontal that is the key here so the angle of depression would be the angle from the horizontal line down to our friend that's this angle right here I'll call it theta so that theta is the angle of depression now an angle of elevation is looking up again from what still from horizontal you're looking up relative to horizontal which means that this would be an angle of elevation looking up at us on the lighthouse our friend down here would be looking at an angle of elevation to get to us I've already used theta so I might need a new variable here except I'm going to use theta again because I'm gonna claim those angles have to be the same think about Y for a second if this line is horizontal and this line is also horizontal those have to be parallel lines then don't they which means that these two angles the angle of elevation here and the angle of depression here those are alternate interior angles theta in blue is the angle of elevation and the most important thing to remember is this relative position again it's looking down from horizontal for an angle depression it's looking up from horizontal for an angle of elevation as long as you remember those horizontal lines then is not a whole lot to trip you up so let's take a look at a few examples of how this would be used of course you're gonna need to know sines cosines and tangents and the inverse trigonometric functions if you need a reminder on any of those I'll put a link in the description to videos on both the regular trig functions and the inverse trig functions for you let's say we're in the woods and we spot a bald eagle up in a trade we measure the angle of elevation to the equal to be 67 degrees I'm going to underline that because that's an important piece of information right 67 degrees is this angle of elevation we walk directly to the tree and determine that we were standing a hundred fifty feet from the tree when we saw the bold Eagle so we were a hundred and fifty feet from the tree when we saw that Eagle the question is how high up was the bald eagle so clearly we need a picture here now there's two kinds of people in the world there's the kind of people that will draw this like this so there's the tree we were over here on the ground when we looked up and saw the Eagle and there's absolutely nothing wrong with that picture we assume that trees grow perpendicular to the ground so we can call that a right angle the angle of elevation was 67 degrees so if this is us over here right then they're telling us that this is a 67-degree angle that we're looking up at the Eagle we were standing 150 feet from the tree and how high up in the tree was the bald eagle let's call that X that mathematically captures everything we need in fact we don't need that person there I got a little carried away if you want to draw a little tree that's totally fine too just don't take like forever on your tree right give it a little bit of detail if you want so this is a basic trigonometric question you think about the angle 67 across from it that is the opposite side of the triangle this side is the adjacent because the side across and the 90 would have been the hypotenuse since we don't know that one will ignore it and we'll just use the opposite and adjacent with those two we can set up a tangent equation we can say that the tangent of 67 is the opposite side x over the adjacent side 150 put that over one and cross multiply X is equal to 150 times the tangent of 67 so grab my calculator and I have approximately 350 3.37 hey let's just round that to 353 there are no rounding instructions so of course whatever instructions are in the problem and that's how told the evil is in the Train notice that in order to answer this question we had to know that the angle of elevation was the angle that we're looking up from horizontal at the Eagle let's take a look at an angle of depression so for this one let's say we're on top of a building and we look down to see our friend on the street I mean if the angle of depression is 57 degrees and the building is 200 feet tall how far away from the bottom of the building is your friend mathematically all we really need to capture what's going on here is a triangle but if you want to you know draw a little building we've got some windows there on a door so of course when we look down at our friend we're looking at an angle of depression but remember what is an angle of depression mean that means we're looking down from horizontal so here's the classic mistake pay careful attention I see a lot of students put that 57 degrees here that is incorrect that is not the angle of depression because you're not looking that angle down from horizontal so the thing you want to be very careful about with an angle of depression problem is always draw in the horizontal line because until you do you can't really see your angle of depression as soon as you put that horizontal line in then you can see that this is the angle that you are in fact looking down from horizontal at your friend and of course that angle is outside of the triangle now right so the triangle here's the right angle the building is 200 feet tall so this side of the triangle is 200 but there's no angle here right this 57 is outside the triangle well that's where those alternate interior angles come in that we talked about before if that angle is 57 so is this because this is horizontal and of course we assumed the ground is horizontal and here we have this set up so I could extract all that information into just a pure triangle it would look like this and we're trying to find how far away from the bottom of the building is our friend that's that distance there we'll simply start off by finding the angle in the triangle across from it is the opposite side this one is the adjacent side across from the right angle would be the hypotenuse and again we have the opposite and adjacent so this is another tangent problem we can say that the tangent of 57 is the opposite 200 over the adjacent X that will give us X times the tangent of 57 equals 200 and here we still need to divide both sides by tangent of 57 so X is actually 200 divided by the tangent of 57 and if you plug that in your calculator you'll get a one twenty nine point eight eight so I'm just gonna go ahead and call that about a hundred and thirty feet again this problem is one that trips a lot of students up because you don't take the time to draw that horizontal line and if you don't put that horizontal line it is very easy to put the 57 degrees here but that will be a completely different triangle and will give you a wrong answer so be careful about that when you're doing an angle of depression always draw that horizontal line one more example because is an extra little wrinkle in this problem we're flying a kite in the park and your friend notices that the angle of elevation from your hand to the kite is 63 degrees that happens all the time right if the kite is a hundred feet off the ground and your hand is five feet off the ground how much string is between you and the kite let's try to build a picture here first if I've got a kite flying up in the air and here's my little kite every kite needs a tail you've ever flown a kite you'll know that that string is not straight it actually curves like this but for our mathematical problem like this they typically assume that the string is straight now here's the thing if our hand is five feet above the ground well then the triangle is actually five feet off the ground too isn't it I here's the triangle that this problem is all about so if I just pull the triangle out I only need ninety five here because the other five feet is the part that's underneath the triangle this this section here right this little rectangle that's five by whatever so I have to take that off the height of the try of the kite the angle of elevation they told us was 63 degrees and of course we're looking for how much string is out that will be that side of the triangle so to solve this we would start on our angle and look across from the triangle there's our opposite side this of course is our hypotenuse because that's the right angle right there so with an opposite and a hypotenuse we can set up a sine equation we can say that the sine of sixty-three is the opposite ninety five over the hypotenuse X cross multiplying we get x times the sine of 63 equals ninety five and then to solve for X we need to divide both sides by the sine of 63 which gives us X about a hundred and six point six so I'm just going to call that about a hundred seven feats of strength so do be careful about a little detail like that if they give you the height of the person or the height of the string for a kite or something like that just realize that your triangles off the ground and you have to account for that in your problem so the last three all we're looking at finding a missing side but we can also use angles of elevation or depression to find missing angles so let's say we're standing on level ground 800 feet from the base of a store now the store has a flag on the top of the building and if it's 50 feet high the building that is what angle of elevation would you need to look in order to see the flag so the building is 50 feet tall that's this side here right the height of the building is 50 and we are down here on the ground and we're gonna look up at this flag and we're going to see it at some angle of elevation we do know that we are 800 feet from the base of the store right so this distance here is 800 feet again we can abstract this into just a triangle also since no information was given about the height of the person or our height in this problem we just assume that that angle is the angle that we are from the ground and solving in this triangle we can say theta is here which means a cross limit is the opposite side of the triangle this of course would be the hypotenuse across in the 90 making the 800 the adjacent side so I have an opposite and an adjacent which means this is going to be a tangent problem the tangent of theta is the opposite over the adjacent side now that's true but it's not helpful because what I really want is to get to that angle theta there so I need to know what angle has a tangent of 50 over 800 so in our calculator going to say well theta is the inverse tangent of 50 over 800 get to that in your calculators you hit the second key and then the tangent button and our calculator will tell us that that angle theta is approximately three point five seven six so let's just go with about three point six degrees take a look at one last problem this time with an angle of depression let's say you're in a hot air balloon flying over Long Island when you spot your house if the balloon is a thousand feet in the air and the distance from your house from you to your house is 10,000 feet what is the angle of depression you're looking at when you see your house and of course we are here in the basket here is distance in the air and then somewhere down here on the ground we look down and we see our house what do we know here the balloon a thousand feet in the air so this side of the triangle would be one thousand the distance from you to your house is 10,000 feet I think carefully where does that distance need to go is this the distance from you to your house now it's this side of the triangle right there's you there's your house so this is the 10,000 of course this is a right angle because that's how we measure height and what we're trying to find is what is the angle of depression that you are looking at when you see your house so remember what an angle of depression is is looking down from horizontal which means the actual angle this question is looking for is this angle here now that angle is not inside of this triangle but that's okay because it's down from horizontal that is the same as this angle because these are alternate interior angles so we're just gonna solve for that angle right there of course we can't abstract this to simply a triangle start at the angle look across from it that's our opposite side this is the hypotenuse so we don't need the other side but that would be the adjacent side with an opposite and hypotenuse this one's going to be a sine problem again so we can say that the sine of theta is the opposite 1,000 over the adjacent 10,000 so this is a true statement but it is not terribly helpful because what I need to know is what's the angle that has a sine that is a thousand over 10,000 that's an easy question to answer as long as we have a calculator handy theta equals inverse sine so sine with that negative 1 of 1,000 over 10,000 to get that again you hit the second button and then the sine button and you'll get that that angle is approximately five point seven three nine and that's pretty much all there is to angles of elevation and angles of depression it's a very simple topic but in my experience it's one that's very easy to trip students up particularly when you have the angle of depression if you just remember to draw that horizontal line and that the angle is from horizontal down then you'll be okay it's putting the angle here that gets you same thing if you put the horizontal line first then you can see your angle of depression in that spot if you put it here which is a very common mistake that's the thing that gets you so just remember angles of elevation aren't looking up from our gentle angles of depression are looking down from horizontal if you enjoyed this video please give it a like subscribe to my channel feel free to leave a comment below and as always have a great day [Music]

[Music] everyone I&#39;m mr. a and today I want to talk with you about an angle of elevation versus an angle of depression this is a very popular topic that comes up when you&#39;re studying trigonometry or geometry and a lot of times those acute those appear together and when we talk about an angle of elevation versus an angle of depression an angle of elevation is when we are looking up at something so think angle of elevation is looking up angle of depression is looking down now up and down a relative terms there&#39;s no such thing as up in an absolute sense right if you imagine right now if you all pointed up you&#39;d be pointing up from the ground but if someone on the opposite side of the world was pointing up they&#39;d be pointing in exactly the opposite direction as you so up and down are always relative to something in this picture here we&#39;ve got us let&#39;s say that&#39;s us on top of this lighthouse and then we have our friend down here on the ground waving to us if we were to look at our friend we would of course I think all agree that we&#39;d be looking down at our friend but looking down from where an angle appellative depression is when we look down from horizontal so it&#39;s not just looking down it&#39;s looking down from horizontal that is the key here so the angle of depression would be the angle from the horizontal line down to our friend that&#39;s this angle right here I&#39;ll call it theta so that theta is the angle of depression now an angle of elevation is looking up again from what still from horizontal you&#39;re looking up relative to horizontal which means that this would be an angle of elevation looking up at us on the lighthouse our friend down here would be looking at an angle of elevation to get to us I&#39;ve already used theta so I might need a new variable here except I&#39;m going to use theta again because I&#39;m gonna claim those angles have to be the same think about Y for a second if this line is horizontal and this line is also horizontal those have to be parallel lines then don&#39;t they which means that these two angles the angle of elevation here and the angle of depression here those are alternate interior angles theta in blue is the angle of elevation and the most important thing to remember is this relative position again it&#39;s looking down from horizontal for an angle depression it&#39;s looking up from horizontal for an angle of elevation as long as you remember those horizontal lines then is not a whole lot to trip you up so let&#39;s take a look at a few examples of how this would be used of course you&#39;re gonna need to know sines cosines and tangents and the inverse trigonometric functions if you need a reminder on any of those I&#39;ll put a link in the description to videos on both the regular trig functions and the inverse trig functions for you let&#39;s say we&#39;re in the woods and we spot a bald eagle up in a trade we measure the angle of elevation to the equal to be 67 degrees I&#39;m going to underline that because that&#39;s an important piece of information right 67 degrees is this angle of elevation we walk directly to the tree and determine that we were standing a hundred fifty feet from the tree when we saw the bold Eagle so we were a hundred and fifty feet from the tree when we saw that Eagle the question is how high up was the bald eagle so clearly we need a picture here now there&#39;s two kinds of people in the world there&#39;s the kind of people that will draw this like this so there&#39;s the tree we were over here on the ground when we looked up and saw the Eagle and there&#39;s absolutely nothing wrong with that picture we assume that trees grow perpendicular to the ground so we can call that a right angle the angle of elevation was 67 degrees so if this is us over here right then they&#39;re telling us that this is a 67-degree angle that we&#39;re looking up at the Eagle we were standing 150 feet from the tree and how high up in the tree was the bald eagle let&#39;s call that X that mathematically captures everything we need in fact we don&#39;t need that person there I got a little carried away if you want to draw a little tree that&#39;s totally fine too just don&#39;t take like forever on your tree right give it a little bit of detail if you want so this is a basic trigonometric question you think about the angle 67 across from it that is the opposite side of the triangle this side is the adjacent because the side across and the 90 would have been the hypotenuse since we don&#39;t know that one will ignore it and we&#39;ll just use the opposite and adjacent with those two we can set up a tangent equation we can say that the tangent of 67 is the opposite side x over the adjacent side 150 put that over one and cross multiply X is equal to 150 times the tangent of 67 so grab my calculator and I have approximately 350 3.37 hey let&#39;s just round that to 353 there are no rounding instructions so of course whatever instructions are in the problem and that&#39;s how told the evil is in the Train notice that in order to answer this question we had to know that the angle of elevation was the angle that we&#39;re looking up from horizontal at the Eagle let&#39;s take a look at an angle of depression so for this one let&#39;s say we&#39;re on top of a building and we look down to see our friend on the street I mean if the angle of depression is 57 degrees and the building is 200 feet tall how far away from the bottom of the building is your friend mathematically all we really need to capture what&#39;s going on here is a triangle but if you want to you know draw a little building we&#39;ve got some windows there on a door so of course when we look down at our friend we&#39;re looking at an angle of depression but remember what is an angle of depression mean that means we&#39;re looking down from horizontal so here&#39;s the classic mistake pay careful attention I see a lot of students put that 57 degrees here that is incorrect that is not the angle of depression because you&#39;re not looking that angle down from horizontal so the thing you want to be very careful about with an angle of depression problem is always draw in the horizontal line because until you do you can&#39;t really see your angle of depression as soon as you put that horizontal line in then you can see that this is the angle that you are in fact looking down from horizontal at your friend and of course that angle is outside of the triangle now right so the triangle here&#39;s the right angle the building is 200 feet tall so this side of the triangle is 200 but there&#39;s no angle here right this 57 is outside the triangle well that&#39;s where those alternate interior angles come in that we talked about before if that angle is 57 so is this because this is horizontal and of course we assumed the ground is horizontal and here we have this set up so I could extract all that information into just a pure triangle it would look like this and we&#39;re trying to find how far away from the bottom of the building is our friend that&#39;s that distance there we&#39;ll simply start off by finding the angle in the triangle across from it is the opposite side this one is the adjacent side across from the right angle would be the hypotenuse and again we have the opposite and adjacent so this is another tangent problem we can say that the tangent of 57 is the opposite 200 over the adjacent X that will give us X times the tangent of 57 equals 200 and here we still need to divide both sides by tangent of 57 so X is actually 200 divided by the tangent of 57 and if you plug that in your calculator you&#39;ll get a one twenty nine point eight eight so I&#39;m just gonna go ahead and call that about a hundred and thirty feet again this problem is one that trips a lot of students up because you don&#39;t take the time to draw that horizontal line and if you don&#39;t put that horizontal line it is very easy to put the 57 degrees here but that will be a completely different triangle and will give you a wrong answer so be careful about that when you&#39;re doing an angle of depression always draw that horizontal line one more example because is an extra little wrinkle in this problem we&#39;re flying a kite in the park and your friend notices that the angle of elevation from your hand to the kite is 63 degrees that happens all the time right if the kite is a hundred feet off the ground and your hand is five feet off the ground how much string is between you and the kite let&#39;s try to build a picture here first if I&#39;ve got a kite flying up in the air and here&#39;s my little kite every kite needs a tail you&#39;ve ever flown a kite you&#39;ll know that that string is not straight it actually curves like this but for our mathematical problem like this they typically assume that the string is straight now here&#39;s the thing if our hand is five feet above the ground well then the triangle is actually five feet off the ground too isn&#39;t it I here&#39;s the triangle that this problem is all about so if I just pull the triangle out I only need ninety five here because the other five feet is the part that&#39;s underneath the triangle this this section here right this little rectangle that&#39;s five by whatever so I have to take that off the height of the try of the kite the angle of elevation they told us was 63 degrees and of course we&#39;re looking for how much string is out that will be that side of the triangle so to solve this we would start on our angle and look across from the triangle there&#39;s our opposite side this of course is our hypotenuse because that&#39;s the right angle right there so with an opposite and a hypotenuse we can set up a sine equation we can say that the sine of sixty-three is the opposite ninety five over the hypotenuse X cross multiplying we get x times the sine of 63 equals ninety five and then to solve for X we need to divide both sides by the sine of 63 which gives us X about a hundred and six point six so I&#39;m just going to call that about a hundred seven feats of strength so do be careful about a little detail like that if they give you the height of the person or the height of the string for a kite or something like that just realize that your triangles off the ground and you have to account for that in your problem so the last three all we&#39;re looking at finding a missing side but we can also use angles of elevation or depression to find missing angles so let&#39;s say we&#39;re standing on level ground 800 feet from the base of a store now the store has a flag on the top of the building and if it&#39;s 50 feet high the building that is what angle of elevation would you need to look in order to see the flag so the building is 50 feet tall that&#39;s this side here right the height of the building is 50 and we are down here on the ground and we&#39;re gonna look up at this flag and we&#39;re going to see it at some angle of elevation we do know that we are 800 feet from the base of the store right so this distance here is 800 feet again we can abstract this into just a triangle also since no information was given about the height of the person or our height in this problem we just assume that that angle is the angle that we are from the ground and solving in this triangle we can say theta is here which means a cross limit is the opposite side of the triangle this of course would be the hypotenuse across in the 90 making the 800 the adjacent side so I have an opposite and an adjacent which means this is going to be a tangent problem the tangent of theta is the opposite over the adjacent side now that&#39;s true but it&#39;s not helpful because what I really want is to get to that angle theta there so I need to know what angle has a tangent of 50 over 800 so in our calculator going to say well theta is the inverse tangent of 50 over 800 get to that in your calculators you hit the second key and then the tangent button and our calculator will tell us that that angle theta is approximately three point five seven six so let&#39;s just go with about three point six degrees take a look at one last problem this time with an angle of depression let&#39;s say you&#39;re in a hot air balloon flying over Long Island when you spot your house if the balloon is a thousand feet in the air and the distance from your house from you to your house is 10,000 feet what is the angle of depression you&#39;re looking at when you see your house and of course we are here in the basket here is distance in the air and then somewhere down here on the ground we look down and we see our house what do we know here the balloon a thousand feet in the air so this side of the triangle would be one thousand the distance from you to your house is 10,000 feet I think carefully where does that distance need to go is this the distance from you to your house now it&#39;s this side of the triangle right there&#39;s you there&#39;s your house so this is the 10,000 of course this is a right angle because that&#39;s how we measure height and what we&#39;re trying to find is what is the angle of depression that you are looking at when you see your house so remember what an angle of depression is is looking down from horizontal which means the actual angle this question is looking for is this angle here now that angle is not inside of this triangle but that&#39;s okay because it&#39;s down from horizontal that is the same as this angle because these are alternate interior angles so we&#39;re just gonna solve for that angle right there of course we can&#39;t abstract this to simply a triangle start at the angle look across from it that&#39;s our opposite side this is the hypotenuse so we don&#39;t need the other side but that would be the adjacent side with an opposite and hypotenuse this one&#39;s going to be a sine problem again so we can say that the sine of theta is the opposite 1,000 over the adjacent 10,000 so this is a true statement but it is not terribly helpful because what I need to know is what&#39;s the angle that has a sine that is a thousand over 10,000 that&#39;s an easy question to answer as long as we have a calculator handy theta equals inverse sine so sine with that negative 1 of 1,000 over 10,000 to get that again you hit the second button and then the sine button and you&#39;ll get that that angle is approximately five point seven three nine and that&#39;s pretty much all there is to angles of elevation and angles of depression it&#39;s a very simple topic but in my experience it&#39;s one that&#39;s very easy to trip students up particularly when you have the angle of depression if you just remember to draw that horizontal line and that the angle is from horizontal down then you&#39;ll be okay it&#39;s putting the angle here that gets you same thing if you put the horizontal line first then you can see your angle of depression in that spot if you put it here which is a very common mistake that&#39;s the thing that gets you so just remember angles of elevation aren&#39;t looking up from our gentle angles of depression are looking down from horizontal if you enjoyed this video please give it a like subscribe to my channel feel free to leave a comment below and as always have a great day [Music]

Transcript for:Angles of Elevation and Depression

Transcript for:
Angles of Elevation and Depression