Trigonometry

• 1. Introduction

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  Trigonometry began with a problem in the sky. In the 3rd century BC, Hellenistic astronomers looking out at the heavens needed a way to turn angles into distances. Out of that need came a branch of mathematics built around the relationships between the angles of a triangle and the lengths of its sides. The right triangle sits at its heart. Its functions tie one acute angle to ratios of edges, and those ratios open a door to almost everything else.

  The field is famous for its identities, equations that hold true for every possible input. People use them to simplify a tangled expression, to find a more useful form, or to solve an equation outright. But how did a tool for tracking stars come to describe sound waves, map continents, and read the inside of a human body? Who first treated this as a discipline of its own, and why did a 16th-century astronomer feel he had to teach it from scratch? The answers run from ancient Sumer to the floating point hardware inside a modern microprocessor.

• 2. From Chords To Sine

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  Sumerian astronomers divided the circle into 360 degrees, and that choice still governs how angles are measured today. They and the later Babylonians studied the ratios of the sides of similar triangles and noticed properties of those ratios. The ancient Nubians used a similar method. None of them turned the observations into a systematic way to find a triangle's sides and angles.

  Euclid and Archimedes, working in the 3rd century BC, studied chords and inscribed angles in circles. They proved theorems equivalent to modern trigonometric formulae, but they presented everything geometrically rather than algebraically. The Greeks worked with chords, not the sine we know.

  Hipparchus, from Nicaea in Asia Minor, gave the first tables of chords in 140 BC. He used them to solve problems in both plane and spherical trigonometry. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy of Alexandria built detailed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy defined his functions through chord length, a small difference from today's sine. To recover the modern sine of an angle, you look up the chord for twice that angle in his table, then divide by two.

  The modern definition of the sine is first attested in the Surya Siddhanta. Its properties were documented further in the 5th century AD by the Indian mathematician and astronomer Aryabhata. Mathematicians in India created the earliest-known tables of values for ratios such as sine. Ptolemy's treatise, meanwhile, stayed in use for astronomical calculation across the next 1200 years, through the medieval Byzantine, Islamic, and later Western European worlds.

• 3. The Islamic Refinement

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  In 830 AD, the Persian mathematician Habash al-Hasib al-Marwazi produced the first table of cotangents. Medieval Islamic mathematicians had taken the Greek and Indian works, translated them, and pushed them further. By the 10th century AD, in the work of the Persian mathematician Abu al-Wafa al-Buzjani, all six trigonometric functions were in use.

  Abu al-Wafa built sine tables in increments of 0.25 degrees, accurate to eight decimal places, along with accurate tables of tangent values. He also made important innovations in spherical trigonometry, the branch concerned with triangles drawn on the surface of a sphere.

  The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right. He was the first to treat it as a subject independent from astronomy, rather than a servant of it. He developed spherical trigonometry into its present form and listed the six distinct cases of a right-angled triangle within it. In his work On the Sector Figure, al-Tusi stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and proved both laws.

• 4. Arrival In Europe

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  De Triangulis, by the 15th-century German mathematician Regiomontanus, is one of the earliest European works on the subject. Knowledge had reached Western Europe through Latin translations of Ptolemy's Greek Almagest and the writings of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi. Regiomontanus was encouraged to write by the Byzantine Greek scholar cardinal Basilios Bessarion, who gave him a copy of the Almagest and with whom he lived for several years. Around the same time, the Cretan scholar George of Trebizond completed another Latin translation of the Almagest from the Greek.

  Nicolaus Copernicus shows how unfamiliar the subject still was in 16th-century northern Europe. He devoted two chapters of De revolutionibus orbium coelestium to explaining its basic concepts. The demands of navigation and the growing need for accurate maps of large areas then turned trigonometry into a major branch of mathematics.

  Bartholomaeus Pitiscus was the first to use the word itself, publishing his Trigonometria in 1595. Gemma Frisius described, for the first time, the method of triangulation still used in surveying today. Leonhard Euler later fully brought complex numbers into the field. The Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century shaped the development of trigonometric series, while Brook Taylor, also in the 18th century, defined the general Taylor series.

• 5. The Six Ratios

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  Two right triangles that share the same acute angle are similar, so their corresponding edge ratios are identical. That single fact is why trigonometric ratios work. Each ratio depends only on one acute angle, which means it behaves as a function of that angle. These are the trigonometric functions.

  The hypotenuse is the side opposite the 90-degree angle. It is the longest side and one of the two sides next to the chosen angle A. The adjacent leg is the other side touching angle A, and the opposite side faces angle A directly. The terms perpendicular and base are sometimes used for the opposite and adjacent sides. Sine is the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent leg to the hypotenuse. Tangent is the ratio of the opposite leg to the adjacent leg.

  The cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent. The cosine, cotangent, and cosecant carry the prefix co- because each is the sine, tangent, or secant of the complementary angle. With these functions, the law of sines and the law of cosines answer virtually any question about an arbitrary triangle. They find the remaining sides and angles once you know two sides and the included angle, two angles and a side, or all three sides.

  Students have long leaned on mnemonics to hold these relationships in mind. SOH-CAH-TOA encodes sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. Some sound the letters out phonetically, close to the word Krakatoa. Others expand them into a sentence such as Some Old Hippie Caught Another Hippie Trippin On Acid.

• 6. Beyond The Triangle

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  The unit circle, a circle of radius 1 centered at the origin, frees these functions from the triangle entirely. The terminal side of an angle placed in standard position meets the circle at a point, and that point's coordinates extend the ratios to every positive and negative argument. From this picture come the familiar values at angles like 30, 45, 60, and 90 degrees.

  Periodicity is what makes the six main functions impossible to invert directly. Because they repeat, they are not one-to-one, so they have no straightforward inverse. Restricting the domain solves the problem and produces the inverse functions: arcsine and arccosine accept inputs between negative 1 and 1, while arctangent and arccotangent accept all real numbers.

  Maclaurin series give another way in. Treated as functions of a real variable, sine and cosine each have a power series representation. Those same definitions extend the functions to complex numbers, where a single formula connects the complex exponential to sine and cosine through e and the imaginary unit i. That complex exponential form turns out to be especially useful.

  Long before calculators, trigonometric functions were among the earliest uses for mathematical tables. Textbooks printed the values, and students learned to look them up and interpolate between entries for more accuracy. Slide rules carried special scales for them. Today scientific calculators have dedicated buttons, usually with a choice of degrees, radians, or gradians. The floating point unit built into the microprocessor chips in most personal computers has instructions for the calculation. Older functions survive in the record too, including the chord, the versine that appeared in the earliest tables, the coversine, the haversine, and the exsecant.

• 7. Where It Reaches

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  Spherical trigonometry has located solar, lunar, and stellar positions for centuries, predicting eclipses and describing the orbits of the planets. In modern astronomy, triangulation measures the distance to nearby stars and underpins satellite navigation systems. The same triangulation, described first by Gemma Frisius, lets land surveyors compute lengths, areas, and the relative angles between objects, and lets geographers measure distances between landmarks.

  Navigation drew on the subject for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances at sea. That work continues through the Global Positioning System and through artificial intelligence for autonomous vehicles.

  Fourier discovered that every continuous, periodic function can be written as an infinite sum of trigonometric functions. Sine and cosine are fundamental to the theory of periodic functions, including those that describe sound and light waves. Even non-periodic functions can be expressed as an integral of sines and cosines through the Fourier transform, which reaches into quantum mechanics and communications. In acoustics and optics, the functions describe sound and light waves and solve boundary- and transmission-related problems. The reach runs further still, into music theory, geodesy, audio synthesis, architecture, electronics, biology, medical imaging such as CT scans and ultrasound, chemistry, number theory and cryptology, seismology, meteorology, oceanography, image compression, phonetics, economics, several branches of engineering, computer graphics, cartography, crystallography, and game development.

• 8. A Web Of Identities

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  Identities are equations true for every possible input, and trigonometry is noted for having many. Those involving only angles are the trigonometric identities. Others, the triangle identities, relate both the sides and the angles of a given triangle.

  The law of sines, also called the sine rule, links a triangle's sides and angles to the area of the triangle and to R, the radius of its circumscribed circle. The law of cosines, the cosine formula, extends the Pythagorean theorem to arbitrary triangles rather than just right ones. The law of tangents, developed by Francois Viete, offers an alternative to the law of cosines when solving for unknown edges, and it gives simpler computations when working with trigonometric tables. Given two sides and the angle between them, the area of a triangle is half the product of those two side lengths and the sine of the included angle.

  The Pythagorean identities tie directly back to the Pythagorean theorem and hold for any value, with two of them derived by dividing the first by a chosen quantity. Euler's formula yields analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i, the same constants that let these functions cross into the complex plane. Beyond these sit the half-angle identities, the angle sum and difference identities, and the product-to-sum identities, the everyday tools for rewriting one trigonometric expression as another.