Proving Pythagorean Triples with Rational Trigonometric Functions

In this article, we explore the properties of right triangles with rational values for sine and cosine of one of their acute angles. Specifically, we delve into two theorems related to these triangles and demonstrate how they can help us understand the formation of Pythagorean triples. Additionally, we revisit Euclid's well-known formula for generating Pythagorean triples and provide a clearer explanation of the underlying principles.

Theorem 1: Rational Sine and Cosine Imply a Pythagorean Triple

Let's begin with Theorem 1, which states that a right triangle with an acute angle alpha such that both cos alpha and sin alpha are rational has a corresponding right triangle that is a Pythagorean triple when the common denominator is cleared.

Proof: Consider a right triangle ABC with the right angle at C. If cos alpha frac{a}{b} and sin alpha frac{c}{d} for natural numbers a, b, c, d, we have:

cos^2 alpha - sin^2 alpha 1

Multiplying both sides by b^2 d^2 gives:

frac{a^2}{b^2} cdot b^2 d^2 - frac{c^2}{d^2} cdot b^2 d^2 b^2 d^2

Simplifying, we obtain:

a^2 d^2 - b^2 c^2 b^2 d^2

This can be rewritten as:

(ad - bc)^2 (bd)^2

Thus, the numbers ad and bd form a Pythagorean triple when bd is the hypotenuse. This confirms Theorem 1.

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Theorem 2: Rational Tangent Function Implies Rational Sine and Cosine of Double Angle

Next, we present Theorem 2, which states that if the tangent of an angle alpha is rational, then both sin 2alpha and cos 2alpha are also rational.

Proof: Assume tan alpha frac{n}{m} for natural numbers m and n. Let z m ni, where alpha arg z. We then have:

tan(2 alpha) tan(2 arg z) tan((arg z)^2) frac{2mn}{m^2 - n^2}

If we set the adjacent, opposite, and hypotenuse sides of the triangle as m^2 - n^2, 2mn, and m^2 n^2 respectively, we can conclude that:

sin(2 alpha) frac{2mn}{m^2 n^2}

cos(2 alpha) frac{m^2 - n^2}{m^2 n^2}

Therefore, both sin(2 alpha) and cos(2 alpha) are rational, proving Theorem 2.

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Rediscovering Euclid’s Formula for Pythagorean Triples

By combining the results of Theorems 1 and 2, we can use Euclid's well-known formula for generating Pythagorean triples:

(m^2 - n^2)^2 (2mn)^2 (m^2 n^2)^2

This formula indicates that for any natural numbers m and n, the numbers m^2 - n^2, 2mn, and m^2 n^2 will form a Pythagorean triple.

In conclusion, we have demonstrated how the rational values of trigonometric functions can help us identify and generate Pythagorean triples. These theorems provide a deeper understanding of the relationship between trigonometric properties and geometric shapes.