To better understand what the Wright Brothers accomplished and how they did it, it is necessary to use some mathematical ideas from trigonometry, the study of triangles. Most people are introduced to trigonometry in high school, but for the elementary and middle school students, or the mathematically-challenged:
There are many complex parts to trigonometry and we aren't going there. We are going to limit ourselves to the very basics which are used in the study of airplanes. If you understand the idea of ratios, one variable divided by another variable, you should be able to understand this page. It contains nothing more than definitions. The words are a bit strange, but the ideas are very powerful as you will see.
Let us begin with some definitions and terminology which we will use on this slide. We start with a right triangle. A right triangle is a three sided figure with one angle equal to 90 degrees. (A 90 degree angle is called a right angle.) We define the side of the triangle opposite from the right angle to be the hypotenuse, "h". It is the longest side of the three sides of the right triangle. The word "hypotenuse" comes from two Greek words meaning "to stretch", since this is the longest side. We pick one of the other two angles and label it b. We don't have to worry about the other angle because the sum of the angles of a triangle is equal to 180 degrees. If we know one angle is 90 degrees, and we know the value of b, we then know that the value of the other angle is 90 - b. There is a side opposite the angle "b" which we will call o for "opposite". The remaining side we will call a for "adjacent", since there are two sides of the triangle which form the angle "b". One is "h" the hypotenuse, and the other is "a" the adjacent. So the three sides of our triangle are "o", "a" and "h", with "a" and "h" forming the angle "b".
We are interested in the relations between the sides and the angles of our right triangle. While the length of any one side of a right triangle is completely arbitrary, the ratio of the sides of a right triangle all depend only on the value of the angle "b". We will illustrate this fact at the bottom of this page.
Let us first define the ratio of the opposite side to the hypotenuse to be the sine of the angle "b" and give it the symbol sin(b).
sin(b) = o / h
The ratio of the adjacent side to the hypotenuse is called the cosine of the angle "b" and given the symbol cos(b). It is called "cosine" because its value is the same as the sine of the other angle in the triangle which is not the right angle.
cos(b) = a / h
Finally, the ratio of the opposite side to the adjacent side is called the tangent of the angle "b" is given the symbol tan(b).
tan(b) = o / a
To demonstrate that the value of the sine, cosine and tangent depends on the angle "b", let's study the three figures at the bottom of the page. We are only going to look at the sin(b) in this example, but there are similar examples for the cosine and tangent. In this example, we have a 7 foot ladder that we are going to lean against a wall. The wall is 7 feet high, and we have drawn white lines on the wall at one foot intervals. The length of the ladder is fixed. If we incline the ladder so that it touches the 6 foot line, the ladder forms an angle of nearly 59 degrees with the ground. The ladder, ground, and wall form a right triangle. The ratio of the height on the wall (o - opposite), to the length of the ladder (h - hypotenuse), is 6/7, which equals roughly .857. This is defined to be the sine of b = 59 degrees. (On another page we will show that if the ladder was twice as long (14 feet), and inclined at the same angle(59 degrees), that it would reach twice as high (12 feet). The ratio stays the same for any right triangle with a 59 degree angle.) Now suppose we incline the 7 foot ladder so that it only reaches the 4 foot line. As shown on the figure, the ladder is now inclined at a lower angle than in the first example. The angle is about 35 degrees, and the ratio of the opposite to the hypotenuse is now 4/7, which equals roughly .571. Decreasing the angle decreases the sine of the angle. As a final check, if we incline the 7 foot ladder so that it only reaches the 2 foot line, the angle decreases to about 17 degrees and the ratio is 2/7, which is about .286. As you can see, for every angle, there is a unique point on the wall that the 7 foot ladder will touch, and it is the same point every time we set the ladder to that angle. Mathematicians call this situation a function.
Since the sine, cosine, and tangent are all functions of the angle "b", we can determine (measure) the ratios once and produce tables of the values of the sine, cosine, and tangent for various values of "b". Later, if we know the value of an angle in a right triangle, the tables will tell us the ratio of the sides of the triangle. If we know the length of any one side, we can solve for the length of the other sides. Or if we know the ratio of any two sides of a right triangle, we can find the value of the angle between the sides. We can use the tables to solve problems. Some examples of problems involving triangles and angles include the descent of a glider, the torque on a hinge, the operation of the Wright brothers' lift and drag balances, and determining the lift to drag ratio for an aircraft.
Here are tables of the sine, cosine, and tangent which you can use to solve problems.
Last Updated Thu, Sep 26 03:10:17 PM EDT 2002