By Jen Kim, Updated Aug 30, 2022
Trigonometry—the study of triangles—has its roots in ancient Egypt and flourished in Greece. It focuses on relationships between sides, angles, and the trigonometric functions that describe them.
The word trigonometry comes from the Greek trigonon (triangle) and metron (measure). The discipline is most often credited to Hipparchus, a 2nd‑century BCE Greek astronomer. By cataloging star positions, he introduced the chord, an early form of the sine function. Later, Ptolemy expanded upon Hipparchus’s work in the Almagest, solidifying trigonometry’s role in celestial navigation.
Perhaps the most famous result in geometry, the Pythagorean theorem states that in a right triangle the squares of the legs sum to the square of the hypotenuse: a² + b² = c². The theorem was first proven by Pythagoras and has since been used worldwide to solve distances, architectural designs, and engineering problems. Integer solutions—known as Pythagorean triples—include (3, 4, 5) and (5, 12, 13).
Six primary functions arise from the ratios of a right triangle’s sides:
The law of sines provides a relationship between any triangle’s sides and angles:
a / sin α = b / sin β = c / sin γ
For example, in triangle ABC let side a = 10 units, angle α = 20°, and angle γ = 50°. Then:
sin 20° / 10 = sin 50° / c
Cross‑multiplying and solving for c gives c = 10 × sin 50° / sin 20° ≈ 22.4 units.
