A Simple Derivation of the Pythagorean Theorem Using the Sine Law
Some four decades ago Dr. Fred Kaempffer, an alumni of Gottingen, and then a professor at a Vancouver University, gave his third year classical physics students a simple question. Is it possible to derive the Pythagorean Theorem … without using circular reasoning!
This very question was asked of him by Werner Heisenberg when Dr. Kamepffer was studying Quantum Mechanics at Gottingen in the 1940’s.
In 1981, of all the students in his third year classical physics class I was the only student to answer his challenge properly. Upon his retirement in 1986, Dr. Kaempffer declared that I was the finest theoretical physicist he ever taught in his five decades of teaching classical and quantum physics at a university level. He gave me autographed copies of his two textbooks and wished me the best!
To derive the Pythagorean Theorem without using circular reasoning I used the sine law.
Say you have a right triangle with sides a, b and c (the hypotenuse) and angles α, β and π/2. This means
By inspection
Similarly
We know the sum of internal angles in a triangle is equal to π.
By inspection, and a fact that can be shown algebraically and numerically,
From whence we see that
Add these two squared terms together
We find that numerically the left hand side is equal to 1 — this can be shown algebraically from first principles using a Taylor series expansion. It can also be shown using Euler’s Treat … which does not rely on trigonometry.
Hence you arrive at the equivalent expression
from which you derive the Pythagorean Theorem.
You in fact haven’t used any trigonometry in this derivation, even starting with the sine law which is a geometric law.
I hear there is some fuss about a ‘new proof’ done by two high school girls that uses the sine law and claims does not use trigonometry.
Tell me is this their new proof? … If it is … you need to know that it is not new at all but has been known for decades.
In essence you can derive the Pythagorean theorem quite easily without using trigonometry.
If you are a competent mathematician you can understand the nuances of this proof.
If you are not, you might start with Euler’s Treat!
