Every time I have asked the question "What is the Pythagorean Theorem?" in a math class, I have always had at least one student answer "a² + b² = c²". It's the easiest formula to remember in math. When I ask the next question, "What are a, b and c?", the answer is often slower to be stated. Sometimes it takes some coaxing of the class to get the meaning, and a student other than the person who correctly stated the formula will correctly state that those letters represent the lengths of the sides of a right triangle, where c is the long side, known as the hypotenuse.
It often happens that we learn a special case of a formula first and the general formula later. The Pythagorean Theorem is a special case of The Law Of Cosines, a rule we can use to find the length of the third side of a triangle if we know the lengths of two sides and the measure of the angle between them. In this picture, side lengths are lower case letters a, b and c, while the opposite angle measures are upper case A, B and C.
The square of the length of the third side is the sum of the squares of the first two sides plus a fudge factor of -2 times the product of the side lengths times the cosine of the angle between those two sides. The cosine function ranges between 1 and -1 as the angle ranges from 0° to 180°. At 90°, cosine is 0, so the fudge factor disappears and we have the nice clean Pythagorean Theorem. If the angle is 0° or 180°, the three sides don't form a triangle, but instead a line segment. At 0°, the third side is the difference of the two sides and at 180°, the third side is the sum. The law of cosines still works in these cases, sometimes called degenerate cases because we get to a situation where we aren't dealing with a legitimate example of the thing we have studied, in this case a triangle. What we get are the algebraic identities
(a - b)² = a² - 2ab + b²
(a + b)² = a² + 2ab + b²
The world is messy and math is clean, but math does a remarkable job of approximating how the world works. One of the reasons math works in the real world is that we understand triangles very well, no matter how they are defined. Problems in physics can often be broken down into parts that relate to the measure of triangles, and there are formulas that can answer nearly any question of measurement of a triangle, relating angles to lengths to areas using just a few elegant ideas.
Bad Signs, Cont.
6 hours ago