 Consider the following situation Shinji while he's traveling is staying on the sixth floor of his hotel building while in his hotel room he looks out the window and observes on a tall building across the street that there are window washers doing their job and he's curious how high are these window washers above me or how high are they above the ground given that Shinji's on the sixth floor of course and so he tries to estimate how high up those window washers are all right so what Shinji first does is he estimates that the the distance between the two buildings is going to be 50 feet which would be the distance between Shinji and the other building as well looking upwards from his window to the window washers he estimates that the angle of elevation would be 80 degrees and then looking down towards the street level at the base of the other building Shinji estimates that the angle of depression would be 50 degrees so using these measurements how high up would the window washers be by Shinji's estimates okay so you can see that with our diagram there's going to be two right triangles coming into play right here because we can assume that these buildings are perpendicular to the ground the first right triangle is going to be based upon Shinji's his angle of elevation so we estimate the angle of elevation was 80 degrees and as he is 50 feet away from the other building we get 50 down here as the adjacent side of this right triangle we need to estimate the h2 here would this would be the height that the window washers are above Shinji and so we could estimate this using a typical tangent ratio notice that tangent of 80 degrees would equal h2 over 50 clearing the dollar as we get h2 is going to equal 50 times tangent of 80 degrees that would be this vertical distance right here but likewise there's a second right triangle in play this one that came from the angle of depression and so if we look at that right triangle Shinji's measurements gave the angle of depression as 50 degrees he also has that the distance here is still 50 feet and so then h1 here would be the opposite side associated to that so by a similar calculation using the tangent ratio we get that h1 is going to equal 50 times tangent of 50 degrees and so the height above the ground of the window washers is going to be h1 plus h2 which this is going to equal 50 times tangent of 80 degrees plus 50 times tangent of 50 degrees for which as 50 is a common factor of both we can factor it out and get 50 times tangent of 80 degrees plus the tangent of 50 degrees now it might be very tempting here to try to add together the angles like just take the take tangent of 130 degrees but tangent doesn't work that way we can't add the angles together we actually have to compute tangent of 80 plus tangent of 50 and so if we need an estimate like Shinji does here uh we should use a calculator so Shinji's just going to pull out his uh his smartphone which has a calculator app on it and so making sure the calculator is in degree mode here Shinji or ourselves would type in tangent of 80 degrees plus tangent of 50 degrees equal and then times that by 50 and then the calculator would tell us that the height of the window washers would be approximately 343.2 feet above the ground