 Trigonometry is useful because it's often easier to measure angles and then compute distances from the trigonometric ratios. One problem is that as a general rule, the trigonometric ratio for a given angle must be found using a calculator or some other computing device. So if you want to find the sine of 40 degrees, you're going to need to punch that into a calculator. More importantly is what you do with those sine and cosine and tangent values. To begin with, we can talk about angles of elevation and depression. And the idea is something like this. Suppose you're walking along and you see something off in the distance. The line of sight is the direct line from the observer to the object. But because this is a line, we can use the line of sight as one side of an angle with the observer at the vertex. If we use a horizontal line as the other, we can define the angle of elevation or the angle of depression. For example, suppose you see an airplane flying at a height of 10 kilometers. The line of sight to the airplane has an angle of elevation of 8 degrees. How far away is the plane from the observer? So let's draw a picture. Now the purpose of drawing a picture is not to exhibit your great artistic talents. But rather it's to keep all of this information organized. First, since the plane is at a height of 10 kilometers, the perpendicular distance from the plane to the ground is 10 kilometers. There's a few other lines that are going to be important. One is a horizontal line. And the other is the line of sight to the plane itself. This is an angle of elevation because we have to look up slightly. And that puts the plane in our field of view. Then we can look straight at the plane. The important thing here is that you have to look at the observer from the outside. So let's change our viewpoint. Let's get rid of some of the scenery. And we'll shift our viewpoint so we can see the plane and the observer simultaneously. Now an important idea in these problems is to look for the right triangle. Now remember height is always measured perpendicular to the ground. Since the plane is at a height of 10 kilometers, this means that the perpendicular distance from the ground to the plane is 10 kilometers. We have the line of sight from the observer to the airplane. And we also have a horizontal line. And this gives us our right triangle. Since the angle of elevation is 8 degrees, we know that the angle between the line of sight and the horizontal has a measure of 8 degrees. If it's not written down, it didn't happen, so let's write that in. And finally the problem asks us to find the distance from the plane to the observer. Well that's this length here, the hypotenuse of our right triangle. We'll give it a creative name. How about X? Now we know this is a trigonometric problem for two reasons. First, it's in the section labeled trigonometry. More importantly, since problems don't come with labels, we see that we have a right triangle, an angle, and a side. And this is exactly the type of setup we have for all trigonometric problems. Now we do need to have the lengths of all three sides. So we have X, the hypotenuse, 10, the side opposite, and we need the side adjacent, so we'll call that B. Because B is the first letter in the word adjacent. So paper is cheap, let's go ahead and just write down our sine, cosine, and tangent. Sine is the opposite over the hypotenuse, cosine is the adjacent over the hypotenuse, and tangent is the opposite over the adjacent. For our angle of 8 degrees, the opposite side is 10, the adjacent is B, and the hypotenuse is X, so we can fill in these values. Now which of these equations do we want to use? Well remember, we want to find this length X, and so tangent isn't relevant because it doesn't even include X. Now in the cosine equation, there's two things we don't know, B and X, and that means we can't solve this for a value of X. On the other hand, in the sine equation, there's only one thing we don't know, and that's what we're looking for. So of these equations, the sine will give us the distance X. So we'll use that equation, the sine of 8 degrees is opposite 10 over hypotenuse X. And let's solve this equation for X. Now as far as the algebra is concerned, 10 over the sine of 8 degrees is a perfectly good answer. If we want to go further, we're going to have to use a calculator or some other device. Here, our angle is measured in degrees, so we'll need to make sure that our calculator is set into degree mode. Or let's take another example. You're standing 15 meters away from a building. The angle of elevation to the top of the building is 78 degrees. How high is the building? So let's draw a picture. So remember the angle of elevation is the angle that the line of sight makes with the horizontal. Now the useful simplification is that unless otherwise indicated, assume the line of sight goes from the ground to the object. Now we'll look for our right triangle. And if it's not written down, it didn't happen. We know the angle of elevation is 78 degrees. We know the distance from the building is 15 meters. We know the distance from the building is 15 meters. And again, unless otherwise indicated, we can assume that that's the length of this line from the building to the vertex of our line of sight angle. We'll want the height of the building, which we'll call x. And since this is a right triangle, the hypotenuse also has a length, which we'll call c. So the sine of 78 is opposite over hypotenuse. And that's cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. Since we want to find x the height of the building, we need to use either sine or tangent. But we don't know c, so we can't use sine. So we'll use tangent. So we have tangent 78 degrees opposite over adjacent. We'll solve for x. We'll use our calculator. Now you should be a little suspicious of a negative answer here. And again, it's important to make sure your calculator is measuring angles the same way you are. Here, our angle is measured in degrees, but our calculator is measuring angles in radians. So we'll need to switch that. And if we round that, we get a height of about 71 meters.