>> In this problem, we'd like to know how far east and how far north the airplane has traveled from this initial point. Now, the, the airplane is traveling for 1.5 hours in that direction at a speed of 550 miles per hour, but how far has the airplane gone? We need a distance here. If we're going to calculate the distances for the, the northerly travel and the easterly travel, you see, and that's what we're doing. We're setting up a right triangle configuration here, and we'd like to know this northerly distance, and let's call it little n, and the easterly distance traveled by the plane. Then we need a value here for this hypotenuse, and that value should be in miles. Well, well, we can easily do that. We know distance is rate times time. We have a rate of 550 per hour and a time of one and a half hours. So we come out here, and we say distance is rate times time. The rate is 550. The time is one and a half hours. Now, once again, notice I, I, I've used the units here just to, to illustrate the idea of the possibility of seeing the cancellation of hours and that our resulting unit of measure needs to be miles here. So, and it's not really a surprise in this problem, but it might be an important matter in, in future problems. OK. Our distance here, then, is 825 miles. Alright. Now, we start to, to calculate the n and the E involved in this right triangle. OK. We're talking about an angle here of 52 degrees. So this is our, the, the perspective from which we're thinking about ratios, and if we're going to calculate n, the northerly distance you see, then from the perspective of this angle, we're talking about the adjacent side and the hypotenuse. So let's see. That adjacent side hypotenuse, that's cosign. So the cosign of 52 degrees is n over 825. So we come out over here, cosign of 52 degrees is n over 825. Multiplying by 825 on both sides, we will calculate the value of n. It'll be 825 times cosign 52 degrees. And when we use the calculator, we find this amount to be approximately, and I'm looking over here because I have it written on the bottom of the board, 508 miles. Now, so this then is about 508 miles. Now when we now arrive at the point of calculating E, are we going to use, you see, from the perspective of this angle, are we going to use the 508 with the E, or are we going to use the 825 with the E? Well, we're certainly not going to use the 508 because it's an approximation. You see, we want to use the given information and what we're looking for as well. So we use 52 degrees, the 825, and the E. Now, from the perspective of this angle, we're talking about opposite side and hypotenuse. That's the sign idea. So the sign, then, of 52 degrees is equal to E over 825. E, then, is 825 sign 52 degrees, and with the calculator, we find E to be approximately 650 miles.