An airplane is descending 225 feet per 1000 feet of horizontal distance covered. What is the cosine of the angle that its path of descent makes with the horizontal?

We find that 0.9756 is the cosine of the angle that the path of descent makes with the horizontal. We need a little drawing. Draw a horizontal line about 4" long and another line about 1" long that goes straight up from the left end of the line. Don't measure. Estimate. You've drawn two legs of a right triangle. Now connect the top of the 1" line with the right end of the 4" line. That's your "path of descent" for the aircraft in the question. It's a hypotenuse. And you've made a scale drawing of the problem. That scale thing is important so that if you mess up, you improve your chances of seeing a problem by applying your math to the scale drawing and "eyeballing it" to see if it seems logical. Make sense? Good. Jump with me. The "angle that its path of descent makes with the horizontal" is that angle on the right. The cosine of that angle is the relationship of the adjacent side (the 1000 foot side) to the hypotenuse, which in our case is 1000 feet over the hypotenuse. But what is the hypotenuse? We drew a right triangle, and the lengths of the sides were 225 and 1000 and the hypotenuse. The sum of the squares of the two sides is the square of the hypotenuse. Let's find the square of the hypotenuse. Our 225 squared is 50,625 Our 1,000 squared is 1,000,000 The sum is 1,050,625 We now have the square of the length of our hypotenuse, so to find the length of the hypotenuse itself we need to find the square root of 1,050,625 which is 1,025 The cosine of the angle of descent is the adjacent side over the hypotenuse, and that's 1000 (the adjacent side) over the 1025 (the hypotenuse), for an answer of 0.9756