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Trigonometric Proof of the
Pythagorean Theorem by Jackson and Johnson (2023) Information
In 2009, Jason Zimba has published a trigonometric proof of the
Pythagorean Theorem discussing the error by Loomis and even pointed out that some Wikipedia editors use this erroneous claim to inappropriately remove content from the Wikipedia article on the
Pythagorean Theorem.[3]: 275
The trigonometric proof by Jackson and Johnson uses a beautiful geometric construction, repeated application of the trigonometric definition of the sine function, and summation of infinite geometric series[4][5][1]. Thus, the Jackson and Johnson proof is different from the proof presented by Zimba.[3]
Trigonometric definition of
sine as a
dimensionless ratio of opposite side to hypotenuse : does not presuppose that right triangles are drawn upon "flat" surface
Euclid's
parallel postulate : introduces the "
flat" geometric space and "
similarity" so that " in a given right triangle (with angles )" can be proved to be equal to the " defined as ratio in any other similar (i.e. "scaled") right triangle (with angles )"
The proof by Jackson and Johnson is valid for almost all right triangles, except for the case when , i.e.
Isosceles Right Triangle.
Case I
For the case when , we can always choose to orient the right triangle so that a<b as shown in the figure.
Step 1: Then we reflect the right triangle to obtain point D. We extend leg BE perpendicular to AB, and extend AD until it crosses BE. From
Euclid's
parallel postulate, it is guaranteed that AD will cross BE because .
Step 2: we apply repeatedly the trigonometric definition of
sine as ratio of opposite side to hypotenuse in order to compute the lengths of the line segments that sum up to AE or BE.
Step 3: we need to sum convergent infinite
geometric series where in order to compute from the large right triangle :
Step 4: we employ the
Law of sines in to find out:
Step 5: upon substituion with and , and after application of basic
arithmetic operations (which we have used also in previous steps), we obtain the
Pythagorean theorem:
.
Case II
For the special case when (hence ), the geometric series does not converge because and also AE does not intersect BE because . However, the proof of the
Pythagorean theorem becomes purely algebraic using the
areas of triangles , and , namely:
, but since , it follows that .
Loomis has dedicated a special section on page 244 in his book in order to state in all caps that there are "NO TRIGONOMETRIC PROOFS"[2]:
NO TRIGONOMETRIC PROOFS
Facing forward the thoughtful reader may raise the question: Are there any proofs based upon the science of trigonometry or analytical geometry?
There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem; because of this theorem we say , etc. Trigonometry is because the Pythagorean Theorem is.
This shows poor understanding by
Elisha Scott Loomis of what trigonometry is. The trigonometric functions sine and cosine are usually defined (
in high school textbooks) as ratio of two sides inside a right triangle and do not necessitate any knowledge of the equality .
The proof by Zimba[3] derives the equality from sine and cosine subtraction formulas. Then, he correctly shows that the
Pythagorean Theorem is a
corollary from the equality . Conversely, the equality can also be derived as a corollary from the
Pythagorean Theorem, which means that the judgement whether a proof is circular or not needs to be made based on understanding of the actual mathematical steps employed.
Circular reasoning occurs only when is assumed to be true without proof, but
circular reasoning does not occur if is first proved and then used for deducing the Pythagorean Theorem.
Nevertheless, a "stubborn" advocate of Loomis may object that Zimba is deriving circularly the Pythagorean Theorem from exactly the equality . Thus, the main point is that Zimba's proof has no whatsoever power over "stubborn" advocates of Loomis.
The novelty and power of the trigonometric proof by Jackson and Johnson is that it never needs to use the equality , hence it achieves what "stubborn" advocates of Loomis claim is "impossible". This indeed justifies the provocatively chosen title of their presentation: "An Impossible Proof Of Pythagoras".[1]
Improved Jackson-Johnston proof based on Einstein's construction Information
The strategy of Jackson-Johnston trigonometric proof can be simplified if combined with
Einstein's construction. The simplification proceeds by noting that instead of doubling the angle to , we can complete it to . Then, we eliminate the need of considering summation of infinite
geometric series and we no longer need the
law of sines.
From the trigonometric definitions of sines inside triangle , we have and .
For the sides in triangle , we have and .
From the triangle , we have:
Substitution of followed by simple algebraic transformations gives:
It is important to note that in contrast to the original construction by Jackson-Johnston which is invalid for , the improved construction that completes the angle to is valid for all possible cases of and .
Difference between trigonometric and analytic definitions of "sine" Information
The high school definition of trigonometric function "sine" as dimensionless ratio of opposite side to
hypotenuse inside a right triangle is curvature-dependent, and is not equivalent to the
analytic definition of the "sine" function using
Taylor series, which is curvature-independent.
Only in flat space, the trigonometric definition of "sine" as ratio of sides
is equivalent to the analytic definition of the "sine" using Taylor series:
In curved space, the two definitions produce different quantities, so they are not equivalent.
Why these two definitions are not equivalent?
Because the geometric expression is curvature-dependent, whereas the algebraic expression is curvature-independent. A visual way to see it, is to draw a right triangle on a
hyperbolic plane and notice that the ratio of the side and hypotenuse has nothing to do with the analytic "sine".
How to verify that these two definitions are not equivalent?
Easy with dimensional analysis: the side length has units of meters, and the hypotenuse has units of meters, so
In contrast, in the analytic expression is dimensionless quantity, i.e., pure number
The above implies that only the expression in which "meters" appear is curvature-dependent. Thus, "trigonometric proofs" of the Pythagorean theorem are not "circular" as long as they use the trigonometric definition of "sine" as ratio of two sides.
Alternative way to see that "there is a difference" in the two definitions is to count the number of
arguments: the trigonometric definition of "sine" is a
multivariable function and has two
arguments, whereas the analytic expression is a single-variable function and has only one
argument.
Why should Wikipedia care?
Because the article should be accessible to high school students.
Is "sine squared" plus "cosine squared" equal to 1 in hyperbolic geometry? Information
There are definitely two non-equivalent definitions of the "sine" and "cosine" functions inside the article on
Pythagorean trigonometric identity that only "accidentally" happen to be both valid due to implicit assumption that the geometry is
flat.
"Trigonometric" sine and cosine in high school are defined as trigonometric ratios
These trigonometric functions are written in bold font for distinction from the subsequent "analytic definition", symbol means "is equal by definition.
The Pythagorean theorem is
"Analytic" functions sine and cosine are defined as Taylor series
Because the sine and cosine series are
absolutely convergent, we can easily compute the squares of these series by multiplication and suitable
rearrangement of the infinite terms
Now summing the latter two analytic expressions, we obtain
The remarkable fact is that the analytic identity, , is always true algebraically regardless of the curvature or flatness of the geometry!
Finally, provided with perfectly explicit and non-ambiguous notation, let us investigate the
Pythagorean theorem and the definitions of trigonometric functions in
absolute geometry in which Euclid's 5th
parallel postulate is not true. If you want a concrete such example, consider
hyperbolic geometry inside which we have the following:
The Pythagorean theorem is false
The Pythagorean identity is false
However the analytic identity is true
Without explicitly stating that there are 2 non-equivalent definitions of sine and cosine, which have different dependence on the curvature of the geometry, it is impossible to answer meaningfully the question: Is "sine squared" plus "cosine squared" equal to 1 in hyperbolic geometry?
References Information
^
abcJackson, Ne'Kiya D. and Johnson, Calcea Rujean (2023).
"An Impossible Proof Of Pythagoras". AMS Special Session on Undergraduate Mathematics and Statistics Research, I: SS30A.{{
cite journal}}: CS1 maint: multiple names: authors list (
link)
^
abLoomis, Elisha Scott (1968).
The Pythagorean Proposition. Washington, D.C.: National Council of Teachers of Mathematics.