Trigonometry: The Mathematics of Triangles and Waves

Trigonometry studies the relationships between the angles and side lengths of triangles. The three primary ratios — sine, cosine, and tangent — relate any angle to ratios of the triangle's sides, summarised by the mnemonic SOH-CAH-TOA. Given any two pieces of information about a right-angled triangle (one side and one angle, or two sides), trigonometry can find the rest. The sine and cosine rules extend this to non-right-angled triangles.

From ancient stargazers to Euler's unit circle

Trigonometry was born from astronomy. The Greek astronomer Hipparchus (around 150 BC) compiled the first known trigonometric table — a table of chord lengths for a circle — to help predict planetary positions. Indian mathematician Aryabhata reformulated the chord into the sine function around 500 AD, a conceptual shift that made the mathematics more powerful. Islamic scholars translated and extended this work, introducing the tangent and cotangent functions. Swiss mathematician Leonhard Euler unified trigonometry with complex numbers in the 18th century through his remarkable formula e^(iπ) + 1 = 0, revealing that sine and cosine are not just geometric ratios but fundamental periodic functions woven through all of mathematics.

Trigonometry in the modern world

GPS systems use trigonometry (and its 3D extensions) to calculate your position from the timing of signals from multiple satellites — a process called trilateration. Architects and engineers use trigonometric ratios to resolve forces into horizontal and vertical components when analysing loads on structures. Sound and light are waves — their behaviour is described mathematically using sine and cosine functions. Music synthesisers generate sounds by combining sine waves at different frequencies. Computer animations interpolate between positions using trigonometric easing functions. Even the spinning of a gyroscope in a phone's compass uses trigonometric decomposition. The triangles in textbooks are just the entry point.