 One very important application of right triangle trigonometry is the ideas of angle of elevation and angles of depression which you see defined here on the screen. An angle measured from the horizontal up is an angle of elevation. So imagine you have like a person who's standing here on the ground and we measure an angle looking up at something. That angle measure right there is an angle of elevation. It's an angle that's going upward from a horizontal line. In contrast, an angle measured from the horizontal down is called an angle of depression. So if there was a horizontal line and the angle's coming down from it, you'd call that an angle of depression. And so these angles of elevation or depression are used all the time with various measurements. So like if we want to measure the height of various objects or the depth of various objects, we can use an angle measurement such as elevation or depression, and then calculate the height or distance of various things using right triangle trigonometry. So consider the following example here. If a 75 foot flagpole casts a shadow of 43 feet long, what is the angle of elevation of the sun from the tip of the shadow? So imagine we have our flagpole right here with our flag waving, and so it casts a shadow on the ground. And that shadow, of course, is a consequence of the sun, you know, beaming there in the sky. And so we want to know what's this angle of elevation? What's the angle of elevation caused by the sun in this situation? So we'll use the measurements we know. So we know that the flagpole is 75 feet tall. We know that we can measure the length of the shadow using, of course, a measuring tape or something. So we see the shadow is 43 feet long. And so as the flagpole is perpendicular to the ground, this forms a right triangle that you can see right here in a much more simplified diagram. We have a right triangle for which we know the adjacent side, we know the opposite side, but we don't know the angle. So we can use a tangent ratio here. Tangent of theta is going to equal opposite over adjacent, which then solving this right triangle, we get that theta is equal to arc tangent, tangent inverse of 75 over 43. Again, this is something we would put into our calculator, and we get an estimate. Make sure your calculator is in degree mode when you do this, of course. We'll get an estimate of 60.17 degrees. So the sun is approximately causing an angle of 60 degrees with its sunlight right there. Here's another example. Imagine that a man is climbing a 200, he climbs 213 meters up on the side of a pyramid as illustrated over here. Now I'm not saying that the pyramid is 213 meters tall. That is the side of the pyramid that the man is hiking up. That is 213 meters tall. At the top of the pyramid, our hiker makes a measurement of the angle of depression. And he finds out that that angle is going to be 52.6 degrees. So using this information, what is the height of the pyramid? That is how far up is the man from the base? That is how tall is this mountain? How tall is this pyramid? That's what we want to think of right here. Now when you look at this picture, there's actually a couple triangles that we've drawn right here. So there's this triangle associated to the piece of the pyramid that the man climbed up. We know this distance and we want to figure out what's the height above the ground going on right here. So we're interested in what's this distance x? How tall is the pyramid? But the man, it would be great if we knew this angle right here. It would be great if we knew this angle theta. Because if we knew this angle theta, then using the sine ratio, we'd get sine of theta equals x over 213, for which then to solve for x, we'll just times both sides by 213, we get x equals 213 times sine of theta. If we knew that angle of elevation, but that angle is inside of the pyramid, which is inside dirt or rock or brick or whatever this thing is made out of. We can't make that measurement, but the thing is the man on the top of the pyramid, he can make a measurement of the angle depression, which is not the angle of elevation, but the related to each other, because of the following principle. Look at this line, which is the height of the pyramid. And then if we take a parallel line that goes through the tip of the pyramid right there, we have two parallel lines. And therefore the slant of the pyramid itself, it acts as a transversal to these parallel lines. And therefore the angle of elevation, which we want, this angle will be an alternate area angle to the angle of depression. So these angles are actually congruent to each other. Or another way of thinking about it is instead of looking at this triangle, we look at this triangle right here, which the hypotenuse of these two right triangles is the same. And these angles are going to be the same. And therefore these line segments are going to be the same as well. And so this is the sort of the convenience of using these alternate area angles. If whether you use an angle of depression or an angle of elevation, it doesn't really matter. The angle measures are going to be the same thing. And so we see that X is going to equal 213 times sine of 52.6 degrees. So again, it doesn't matter whether you use this triangle right here with an angle of elevation, nor does it matter if you use this triangle right here and use an angle of depression. The two are going to be the same angle. It's just relative. Which direction are we looking at? That is the difference between them. And so then put into your calculator sine of 52.6 degrees. Make sure in the degree mode, plug that in. And then whatever you plug that in times that by 213 in your calculator, don't clear out the memory. And you're going to see that the height of the pyramid is approximately 169 meters. From there, we can measure like the slope. How steep is it? What's the grade of it? And this type of this type of calculations using right triangles is used all the time in various surveying activities.