Hellenistic skies first forced triangles into numbers
Trigonometry—from Greek words for triangle and measure—studies how angles and side lengths talk to each other. It grew in the Hellenistic world in the third century BC for astronomy: Greeks computed chords, while Indian mathematicians built the earliest-known tables of ratios such as sine. Surveyors, navigators, and celestial mechanics still live on those ratios.
In a right triangle the ratios depend only on one acute angle, because any two right triangles sharing that angle are similar. Sine is opposite over hypotenuse; cosine is adjacent over hypotenuse; tangent is opposite over adjacent—also sine divided by cosine. Cosecant, secant, and cotangent invert those three. The “co-” names mark complementary angles: cosine is sine of the complement, and likewise for the others. Mnemonics such as SOH-CAH-TOA, sometimes stretched into comic sentences, help students keep the primary three straight.
With the law of sines and the law of cosines, two sides and the included angle, two angles and a side, or three sides suffice to finish any triangle. The unit circle of radius 1 extends the same ratios to all positive and negative angles. Because the six main functions are periodic they fail to be one-to-one, so inverses need restricted domains. Maclaurin series write sine and cosine as infinite polynomials, and Euler’s formula binds the complex exponential to cosine and sine of the imaginary part, welding trigonometry to analysis.
Before calculators, textbooks printed tables and taught interpolation; slide rules carried special trigonometric scales. Modern scientific calculators offer sin, cos, tan and inverses in degrees, radians, or sometimes gradians, and floating-point hardware in personal-computer chips includes trig instructions. Older companions—chord, versine, coversine, haversine, exsecant—filled early tables and navigation formulas even though they are rare today.
Spherical trigonometry, for centuries, located the Sun, Moon, and stars, forecast eclipses, and described planetary orbits. Identities remain the craft’s everyday tools: rewrite, simplify, or solve.
Source: Trigonometry
