 In a previous screencast, we discussed the concept of solving a right triangle. In this screencast, what we're going to do is start with a problem that is given to us verbally and with a diagram, translate that problem into a right triangle problem, then we'll solve the right triangle problem, which will also solve the original problem that we were given. So, here's kind of a summary of how we work with a right triangle. This is a general thing about solving any triangle, but what we are going to be discussing here applies only to right triangles. So in other words, we can only use this process when we know we have a right triangle. And the idea here is that for any triangle, the way we're working with it, there are six quantities involved. There are the three sides, the lengths of the three sides, and the measures of the three angles. And of course, in this situation, one of the angles has to be a right angle. So, as we said here, these types of problems can then be solved using the trigonometric ratios and the Pythagorean theorem. And in addition, we know that the sum of the two angles other than the right angle have to add up to 90 degrees. So, here's kind of what we have for this again. If we label an angle theta, so we're working with this angle here. And in relation to the angle theta, this side over here is the side opposite. This is the adjacent side, and of course, we have the hypotenuse. So, here we have our sine, cosine, and tangent. And which one we decide to use depends on what information we're given. The other thing that we know, again, is the Pythagorean theorem, and in sort of using this notation, we could write this as, say, the adjacent side squared, technically the length of the adjacent side, plus the length of the opposite side squared equals the length of the hypotenuse squared. And also, if we label this angle, say, as phi, the Greek letter phi, we would also know that theta plus phi has to be 90 degrees. So, in other words, if we know one of the two angles, we can determine the other angle. So, here's our problem, and we can read this. But as you read this, again, notice that we're measuring the distance from the top edge of a bank down to the bottom edge by the creek there. So, that distance here is 41.8 feet, and surveyors would also have an instrument that they would be able to measure this angle with the horizontal. So, again, we have, the other thing we have to assume is that when we have a horizontal distance and in a vertical distance, in between those two form a right angle. And so, with that, then, we can determine the horizontal distance from the top of the bank to the edge of the creek and then determine, in effect, the vertical distance, how far the creek is below the level of the surrounding land. So, as I said, the first thing we want to do is get a right triangle out of this, and it's pretty well set up. We have to introduce some notation to help us out. So, we will let H be the horizontal distance from the top of the bank to the edge of the creek and let V be the vertical distance. And, again, what we have here is a right angle right there. So, now, again, our angle that we're focusing on is right here. So, as we look at this, V is the opposite side and H is the adjacent side. And we're given the value of the hypotenuse. So, one thing we can say, is if we would look at H over 41.8, what we're looking at is the ratio of the adjacent side to the hypotenuse. And so, that's equal to the cosine of the angle, which is 28.2 degrees. So, now, we have a simple equation, H over 41.8 equals cosine of 28.2. We've multiplied both sides of that equation by 41.8 and we get this for H. So, it's 41.8 times the cosine of 28.2 degrees. And if we take out our calculator now, making sure our calculator is in degree mode, we will find that H is approximately 36.8 feet. And I rounded that off to the nearest tenth of a foot because our other measurement was basically given in tenth of a foot, 41.8 feet. And now, to get that vertical distance, it's effectively the same type of computation. The only difference is if we look at V over 41.8, we have the opposite side over the hypotenuse. And so, we can say V over 41.8 is the sine of 28.2 degrees. Or V equals 41.8 times the sine of 28.2 degrees. And again, this is where we take out our calculator and get an approximation for that. And again, rounding to the nearest tenth, we get 19.8 feet. And in effect now, we've solved a given problem about the only thing we haven't determined in this right triangle is the measure of this angle, but that's not required for this problem and what we could get it if we wanted. So, in summary, answering the question, using a complete sentence, the horizontal distance from the top of the bank to the edge of the creek is 36.8 feet and the creek is 19.8 feet below the level of the surrounding land. One last thing that's always nice to be able to do is to get some kind of check for your work. And in this case, we'll use a Pythagorean theorem. In some problems, we will use it to find certain values, but in this case, we can use it as a check. And in particular, we can have 36.8 squared plus 19.8 squared. And again, if we do that computation on our calculator, we get 1746.28. And of course, that's supposed to be equal to the hypotenuse squared. Now, one thing that you have to be aware of at this point is that because we rounded the values, this check will not be exact. And in fact, if we do 41.8 squared, we do get 1747.24. And that we have to accept as being close enough. If we wanted to get a more precise check, we would have had to use more precise answers for the 36.8 and the 19.8. In other words, maybe use the 3, 4, or 5-digit approximation. Our check would have been much better. So, there's one problem that we solved using right triangles. And the hope is that understanding the process for solving this problem will also help you solve other right triangle problems. Thanks for watching.