MATHEMATICAL PERSPECTIVES
A Trigonometric Fallacy to Provide Deeper Understanding in Mathematics
By Dr. Alfred Posamentier
We know that there are more than 400 proofs of the Pythagorean theorem, yet none of them uses trigonometry. Students are often asked why can we not use trigonometry to prove the Pythagorean theorem? The answer is very simple. The basis for trigonometry is the Pythagorean theorem, and therefore, it would be fallacious reasoning to use trigonometry to prove a theorem on which it is based.
In trigonometry, the Pythagorean theorem often manifests itself as cos2 x + sin2 x = 1. From this we can show that 4 = 0. It is to be assumed that you know this cannot be true. So it is up to you to find the fallacy as it is made. If you don’t, we’ll expose it at the end of the unit.
The Pythagorean identity can be written as cos2 x = 1 – sin2 x. If we take the square root of each side of this equation, we get: cos x = (1 – sin2 x)1/2.
We will add 1 to each side of the equation to get: 1 + cos x = 1 + (1 – sin2 x)1/2.
Now we square both sides: (1 + cos x)2 = [1 + (1-sin2 x)1/2]2.
Let us now see what happens when x = 180°. Cos 180° = -1, and sin 180° = 0.
Substituting into the above equation gives us:
(1 – 1)2 = [1 + (1-0)1/2]2
Then 0 = (1+1)2 = 4.
Since 0 *******4, there must be some error. Where is it? Here is a hint:
When x2 = p2, then x = +p, and x = –p. The problem situation may call for one or both of these values. Yet sometimes one of them won’t work. Look at the step where we took the square root of both sides of an equation. There lies the culprit!
It is topics like this and presented in an enthusiastic fashion that more math teachers should use to enhance their teaching and generate greater interest in mathematics. We must get away from “teaching to the test.” #
