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Welcome to trigonometry. You're making a triangle. So let's make one. Picture a line on flat ground that is 100 feet long. At the "left" end, we'll put a stick straight into the ground and standing perfectly upright. At the other (the "right") end, we'll put our "angle measurer" or protractor, and (somehow) sight along it at 1o to our stick. Our assistant will mark the stick at the place where our line intersects it with out 1o angle of arc. Guess what! We now have a right triangle. The right angle is the one between the stick and the ground. In our picture, it is on the left. We have our 1o angle at the measured end (the right of our picture), and we're ready to solve. So let's solve! In a right triangle, we have a right angle and a side, called the hypotenuse, that is opposite the right angle. That leaves us the two other sides. We know our angles because we measured the 1o angle, we have the right, or 90o angle (at our stick and the ground) and we have the last angle at the top of the stick where the line we sighted in intersects the stick. It's 89o because we know the sum of the interior angles in all triangles is 180o, and we have 90o and 1o in our other angles. Simple math. On to the solution. There are a set of rules associated with all right triangles. They are always true. They are the trigonometric rules, or what we call trigonometric functions. The trig functions are ratios. It's that simple. They are numbers that have no units attached to them. There are three of them and they each have a reciprocal, so there are six if you want to look at them that way. The "big 3" are the sine, cosine and tangent. Each speaks to a different ratio. And we are going to use one of them here - the tangent. The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Note that any reference to a side in a trig function is made to a side that is not the hypotenuse. If we are making reference to the hypotenuse, we'll say "hypotenuse" and not "side" in our ratio, okay? Let's jump on this. The tangent of 1o is the ratio of the length of the "short" side (that side from the ground up along our stick to the mark we made when we measured) to the "long" side (which is the 100 feet we marked on the ground). We get our calculator and find the tangent (tan) of 1o and we will get 0.017455 for our answer. That's a pure number, now. Not feet or inches or anything else, okay? So the ratio of the length of our "mystery" side to the 100 feet of our known and measured side is 0.017455 (per our trig functions). That means that the length of the side we want to find (which we will designate as x) divided by our 100 feet will equal 0.017455 and it will look like this: x / 100 feet = 0.017455 To find x, which is the length of the side we want to know, we have to multiply both sides of our equation by 100 feet. (x / 100 feet ) 100 feet = (0.017455) 100 feet x = (0.017455 x 100) feet = 1.7455 feet Answer: 1.7455 feet (or just a hint short of 19 inches)
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