The following proof of the Pythagorean theorem using trigonometry was discovered (or is the proper word invented?) by David Houston, an eighth-grade student from Sterling Heights, Michigan, who comes to Oakland University to take mathematics courses and to chat with some of our faculty members on a regular basis. Bright students in a trigonometry class or a geometry class should be able to follow it.
If we define sinθ for any acute angle θ as the ratio of the opposite side to the hypotenuse in a right triangle with angle θ, then the area formula
To establish the Pythagorean theorem, we want to prove that
Without loss of generality, we assume that
In the figure on the left, we see that on the one hand, the total area is
ab = c² sinθ / 2.
The figure on the right is formed by reflecting the original triangle about the line through the right-angle vertex, parallel to the hypotenuse. Since each copy of the right triangle clearly occupies half of its half of the large rectangle, the total area is 2ab. On the other hand, if we add the areas of the four triangles in this figure, we see that the area ab (from the two copies of the original right triangle) plus
ab = (a² + b²) sinθ / 2.
The Pythagorean formula then follows immediately from the two displayed equations.
As I mentioned in the introduction to the Pythagorean theorem page, E. Loomis expressed an opinion that the Pythagorean theorem does not admit a trigonometric proof. That opinion is no doubt shared by the majority of mathematicians. The reason for this is the fact that
The definitions of trigonometric functions are based on the theory of proportion and similarity. For example, all right triangles with the same angle α are similar and, for similar triangles, the ratios of two corresponding sides are equal; in particular, the ratio of a leg to the hypotenuse is a function of the adjacent (or the opposite) acute angle. For the adjacent angle the function is called cosine, for the opposite angle it is called sine. As long as a proof of the Pythagorean identity contains only the definitions of trigonometric functions, one may reasonably claim that the use of trigonometry is entirely spurious. The functions can be simply replaced by the ratios, leading to a plain algebraic proof. A remark to this effect has been made at the end of Proof #6.
David Houston's proof makes use of trigonometry twice. First, the proof depends on the trigonometric formula for the area of a triangle
We might have proceeded as follows.
The theory of similarity and proportion allows us to make an observation: let x, y be the sides of a triangle with the included angle θ. Let h be the length of the altitude to side x. Then the ratio h/y is the function of (depends only on) angle θ. Let's denote this function as F: h/y = F(θ).
The above diagram implies
David's configuration is obtained by reflecting the above in the lowest orange line. The argument that also employs similarity of triangles, proportions and areas could be extended to David's configuration.
However, David's argument is different and, to boot, the trigonometric formula for the area of a triangle plays such a prominent role in the proof that replacing sine with an auxiliary function F seems (to me at least) a rather artificial device. In a hindsight, one was indeed able to do away with trigonometry thus confirming Loomis' view. On the other hand, there are grounds for a reasonable doubt whether anybody without the knowledge of trigonometry would have been able to come up with such a nice and simple proof as David's.
Note: Luc Gheysens came up with a modification of David Houston's proof that leads to a shorter derivation.
- I. M. Gelfand, M. Saul, Trigonometry, Birkhäuser, 2001
- J. Grossman, a reader's letter in Mathematics Teacher, v. 87, n. 1, January 1994, NCTM
- E. S. Loomis, The Pythagorean Proposition, NCTM, 1968