The Genesis of Sine and Cosine: Before and Beyond the Pythagorean Theorem

Are sine (sin) and cosine (cos) a direct result of the Pythagorean theorem, or do they exist independently before its formal statement? The relationship between the two and the theorem is intricate and multifaceted, offering deeper insights into the history and applications of these fundamental trigonometric functions.

Origins of Trigonometry

The concepts of sine and cosine were developed in ancient civilizations such as the Greeks and Indians, long before the formalization of the Pythagorean theorem. The earliest known use of sine dates back to Indian mathematicians like Aryabhata around 500 CE. The Pythagorean theorem, on the other hand, was known to the Babylonians and Greeks, but the definitions of sine and cosine were established separately and evolved over time.

Pythagorean Theorem and Beyond

The Pythagorean theorem, stating that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2 b2 c2), was indeed known to the ancient civilizations, but it did not directly give rise to the sine and cosine functions.

Unit Circle and Trigonometric Functions

Sine and cosine can be defined using the unit circle, where a circle of radius 1 is centered at the origin of a coordinate plane. For an angle θ, cos θ represents the x-coordinate, while sin θ represents the y-coordinate. This relationship is encapsulated in the Pythagorean identity: sin2θ cos2θ 1, which arises from the Pythagorean theorem applied to the unit circle.

Trigonometry through Similarity and Ratios

Trigonometry fundamentally rests on the concept of ratios of sides in similar triangles. The ratios a/b, a/c, and b/c are uniquely determined by the three angles A, B, and C. For example, in an isosceles triangle with a vertex angle A, the base angles B and C are each 90° - A/2. In a right-angled triangle, the ratio a/b can be expressed as the sine of the angle B.

Any triangle can be dissected into three isosceles triangles by joining the circumcenter to the vertices, allowing the ratios to be expressed in terms of the angles. This approach was the original Greek version of trigonometry. Aryabhatta's method of breaking an isosceles triangle into two right-angled triangles through a median/altitude further refined these concepts.

Historical Context

The isosceles triangle's chord (2 sin A/2) is a key function in trigonometry, but it lacks a simple relationship with the Pythagorean theorem due to its nature. Sin and cos, however, are more closely linked to right-angled triangles, making them more integral to the theorem's application.

Conclusion

In summary, while sine and cosine existed as mathematical concepts before the Pythagorean theorem was formally stated, they are intricately connected to it through their definitions and relationships in the context of right triangles and the unit circle. The historical development of trigonometry shows a rich interplay between these concepts, reflecting the deep interconnections in mathematics.