# History of Trigonometry: Who Invented Sine and Cosine?

*Science & Technology · Priya Nair · 2026-08-09*

## TL;DR

The history of trigonometry begins with Greek astronomers like Hipparchus in the 2nd century BCE, who used chords to study the heavens. Indian mathematicians later refined these ideas into half-chords, the direct ancestors of sine. Islamic scholars preserved and expanded the field, and European mathematicians eventually named the functions sine and cosine.

## Key facts

- Hipparchus of Nicaea, working around 150 BCE, created the first known trigonometric table using chords to solve astronomical problems.
- Indian mathematician Aryabhata, around 500 CE, developed the concept of the half-chord, which became the foundation of the modern sine function.
- Islamic scholars such as Al-Battani and Abu'l-Wafa refined trigonometric methods during the 9th and 10th centuries, introducing the tangent and secant functions.
- The term "sine" originated from a Latin mistranslation of the Arabic word "jiba," which itself derived from the Sanskrit term for half-chord.
- Edmund Gunter, an English mathematician, coined the term "cosine" in 1620 as an abbreviation for "complementary sine."

## The Sky's Secret Code: How Trigonometry Was Born

Long before anyone drew a right triangle on a chalkboard, the history of trigonometry began with people staring up at the night sky. The Egyptians, Babylonians, Indians, and Chinese all developed early astronomical methods for tracking stars, building calendars, and navigating their world. They needed to measure distances they couldn't reach and predict movements they could only observe, and angles became their most powerful tool.

These ancient civilizations divided the sky into sections, recorded star positions, and created remarkably accurate calendars from their observations. But it was the ancient Greek mathematicians who transformed this practical knowledge into systematic geometry. In the 2nd century BC, Hipparchus took a crucial step by creating a table that linked angles to chord lengths on a circle. According to Theon of Alexandria, Hipparchus wrote a 12-book work on this subject, though the original text has been lost to time, so we cannot be certain exactly how his table was organized or what intervals it contained.

This chord table, built from circle geometry rather than triangles, became the direct forerunner of our modern sine tables. The Greeks used the full chord corresponding to an angle, while later Indian mathematicians would shift to using half the chord, a change that would eventually give birth to the sine concept itself. The sky, it turns out, was the original trigonometry textbook.

## From Half-Chords to Jya: The Indian Turn That Gave Us Sine

The Greeks measured whole chords across a circle, but it was Indian mathematicians who sliced those chords in half and, in doing so, quietly invented the sine. India had a deep, independent astronomical tradition, and that tradition demanded better tools for tracking the heavens.

Picture a chord drawn across a circle. Drop a perpendicular line from the circle's center to that chord, and the perpendicular splits the chord into two equal halves, creating two neat right triangles. Those right triangles made the math dramatically simpler. Indian astronomers realized they could work with just one of those halves instead of wrestling with the entire chord, and they built their tables around this smaller, handier length.

The terminology they used tells the story beautifully. In Sanskrit, the full chord was called *samastajya*, where *samasta* means "full" and *jya* means "bowstring." The half-chord was *ardhajya*, later shortened through everyday use to simply *jya*. That shortened word, *jya*, is exactly what we now know as the sine. The shift from full chord to half-chord was the pivotal turn in the history of trigonometry, the moment the modern sine concept was born.

## The Islamic Golden Age: How Scholars Built Trigonometry Into a Science

By the 8th century, Baghdad had become the beating heart of a translation movement that would reshape mathematics forever. Greek, Indian, and Persian astronomical works poured into the city's libraries and were rendered into Arabic, not as museum pieces but as working tools. The scholars who received these texts did something remarkable: they took the half-chord tables from India and the chord geometry from Greece and pushed both much further.

Al-Battani, working across the 9th and 10th centuries, turned sine and tangent into everyday instruments for astronomical calculation. Born in Harran around 858, he used these functions systematically to track the Sun and Moon, producing tables and relations that made predictions noticeably sharper than anything that had come before. His precision mattered because prayer times, calendar dates, and the visibility of the new moon all depended on getting the numbers right.

Then came Abu al-Wafa al-Buzjani and Biruni, two mathematicians who advanced both plane and spherical trigonometry in the 10th and 11th centuries. Spherical trigonometry was no abstract luxury: it let astronomers calculate where celestial bodies would appear, which direction cities lay from one another, and crucially, the qibla, the exact direction of Mecca that Muslims needed for daily prayer. Biruni, born in 973 in the region of modern Uzbekistan, went further and used trigonometric methods to compute the Earth's radius and geographical distances, measurements that required both mathematical skill and careful observation.

The real turning point arrived in the 13th century with Nasir al-Din al-Tusi. In his *Treatise on the Quadrilateral*, al-Tusi gathered plane and spherical trigonometric relations into a systematic whole and, decisively, treated them as a field of mathematics in their own right rather than a servant of astronomy. That move effectively gave trigonometry its independence as a discipline.

One naming curiosity lingered through all this: cosine had no separate name for a long time. Mathematicians simply referred to it as *sinus complementi*, Latin for "sine of the complementary angle." The term cosine, as we know it, emerged later from that phrase, though the exact attribution remains uncertain.

## The Curious Birth of the Word 'Cosine'

For a long stretch of mathematical history, nobody felt the need for a separate word for cosine at all. Mathematicians simply described it the long way, in Latin, as *sinus complementi*, which translates to "the sine of the complementary angle." That clunky phrase worked fine for centuries, until an English mathematician named Edmund Gunter reportedly got tired of it and chopped it down.

The story most often told goes like this: in 1620, Gunter abbreviated *sinus complementi* to *co-sinus*. From there, the term supposedly slipped into *cosinus* and eventually settled into the familiar *cosine* we use today. It's a tidy little origin tale, the kind that makes the history of trigonometry feel neat and linear.

But here's where the story gets murky. Contrary to the popular version, the exact year and the attribution to Gunter are actually uncertain. Records don't clearly pin the coinage to 1620, and historians can't say with confidence that Gunter was the one who first used the abbreviation. What does seem clear is that the concept itself emerged from that practical habit of describing cosine in relation to the sine of the complementary angle, a relationship that had been in use long before anyone gave it a short name.

So while the word "cosine" feels like a simple, modern label, its birth is more of a slow evolution than a single moment of invention. The idea came first, the Latin phrase came second, and the shortened name came whenever it came, likely through the everyday shorthand of working mathematicians rather than one dramatic announcement.

## Why Trigonometry Still Runs the World

That ancient tool built for star-gazing and pyramid-measuring? It silently powers your phone, your car, and the bridge you drive across. Trigonometry never retired. It simply traded clay tablets for microchips, and today it handles everything from surveying land you cannot walk across to keeping the electrical grid humming in perfect rhythm.

Surveyors are among its most devoted daily users. When a river or a ravine blocks the path between two points, the distance cannot be stretched across with a tape measure. Instead, surveyors stand at one bank, sight the target on the other, measure the angle, and let a sine or cosine calculation deliver the missing length. The same logic lets engineers split a diagonal force into its horizontal and vertical parts, which is exactly how they decide how thick a bridge cable must be or how a building will sway in the wind. Road designers lean on the same ratios to shape safe curves and gentle gradients.

Your GPS receiver is essentially a trigonometry problem in orbit. It takes timing signals from satellites and, using geometric and trigonometric principles, triangulates your position on the planet's surface. Without those angle-to-length relationships, the map pin that marks your café would be a guess, not a location.

Then there is the world of motion. A pendulum swinging, ocean waves rolling onto shore, the vibration of a guitar string, and the alternating current that powers your outlets are all periodic, repeating phenomena. Sine and cosine functions model each one with striking accuracy. Engineers and physicists lean on these functions constantly because they translate a repeating rhythm into numbers they can calculate with.

Perhaps the deepest idea is that sine waves are treated as the natural oscillation wave. Any waveform you can imagine, from the music streaming through your headphones to the Wi-Fi signal carrying this page to your screen, can be built by stacking sine waves. That single insight, born from ancient chord tables, is the quiet engine behind modern signal processing and digital communication. The history of trigonometry gave us a language, and we still speak it fluently in every device we touch.


## Frequently asked questions

**Who invented trigonometry?**

No single person invented trigonometry. The field grew gradually through contributions from multiple civilizations over more than a thousand years. Greek astronomer Hipparchus created chord tables in the 2nd century BC, Indian mathematicians developed the half-chord concept that became sine, and Islamic scholar Nasir al-Din al-Tusi established trigonometry as an independent mathematical field in the 13th century. The story of trigonometry is really a relay race across centuries. Greek astronomers working around the 2nd century BC started mapping the relationship between angles and chord lengths on circles. Indian mathematicians then shifted focus to the half-chord, a concept that would evolve into the modern sine function. When these works reached the Islamic world through translation movements starting in the 8th century, scholars refined and expanded them. By the 13th century, al-Tusi's Treatise on the Quadrilateral pulled everything together into a standalone mathematical discipline, no longer merely a helper for astronomy.

**What does the word sine mean and where does it come from?**

The word "sine" traces back to the Sanskrit term "jya," meaning half-chord. Indian mathematicians originally used "ardhajya" for half-chord, later shortened to "jya." This term traveled through Arabic translations into Latin, where it eventually became "sinus," the Latin word for curve or fold. Indian astronomers working with celestial calculations found that half-chords were more practical than full chords. They documented these values in tables, creating what we now recognize as early sine tables. The terminology journey is fascinating: "ardhajya" became "jya," which Arabic translators rendered as "jiba," and Latin scholars later transformed into "sinus." The word stuck, even though its original geometric meaning of a bowstring or chord had faded from memory.

**Why is cosine called cosine?**

Cosine comes from the Latin phrase "sinus complementi," meaning "sine of the complementary angle." English mathematician Edmund Gunter reportedly shortened this to "co-sinus" in the 17th century, though the exact year and attribution remain uncertain. The term eventually evolved into "cosinus" and then "cosine." For centuries, mathematicians simply described cosine as "the sine of the complementary angle" without giving it its own name. This made sense because the cosine of an angle equals the sine of its complement, the angle that adds up to 90 degrees. The abbreviation to "co-sinus" was a practical shorthand that caught on, giving us the compact term we use today.

**How did Islamic scholars contribute to trigonometry?**

Islamic scholars translated and refined existing Greek and Indian works, introduced systematic use of sine and tangent through al-Battani in the 9th and 10th centuries, advanced spherical trigonometry through Abu al-Wafa and Biruni, and finally established trigonometry as a separate field through al-Tusi in the 13th century. The translation movement centered in Baghdad from the 8th century brought Greek, Indian, and Persian astronomical works into Arabic. Al-Battani used sine and tangent systematically in his astronomical calculations, improving the precision of solar and lunar motion predictions. Abu al-Wafa and Biruni pushed spherical trigonometry forward, which proved essential for calculating celestial positions and directions between cities. Biruni even used trigonometric methods to compute the Earth's radius and geographical distances. Al-Tusi's 13th-century work marked the culmination, reorganizing trigonometric relationships into a coherent theory independent of astronomy.

**What is the difference between sine and cosine?**

In a right triangle, sine of an angle equals the ratio of the opposite side to the hypotenuse. Cosine equals the ratio of the adjacent side to the hypotenuse. The two functions are related through complementary angles: the sine of an angle equals the cosine of its complement, meaning sin(A) = cos(90° - A). This complementary relationship explains why the names are so similar. When you know one value, you can quickly find the other. The practical difference comes down to which sides of the triangle you are comparing. Engineers and physicists choose between sine and cosine depending on which direction they need to analyze, whether that is vertical or horizontal components of a force, or the height versus the horizontal distance of a slope.

**How is trigonometry used in modern GPS technology?**

GPS systems use satellite signals and geometric principles to calculate a receiver's position on Earth. The process, called trilateration, involves measuring distances to multiple satellites and using those distances to determine the receiver's location through geometric calculations that rely on trigonometric relationships. When your phone or car navigator receives signals from GPS satellites, it measures how long each signal took to arrive. That time translates into a distance from each satellite. With distances to several satellites known, the receiver can calculate its position through geometric triangulation. Trigonometry handles the angular relationships and coordinate transformations needed to convert satellite-based measurements into a precise location on the curved surface of the Earth.

**What is a sine wave and why is it important?**

A sine wave is a smooth, periodic oscillation described by the sine function. It is fundamental to physics, engineering, and signal processing because any complex waveform can be constructed from combinations of sine waves. This makes sine waves the building blocks of all periodic motion and signals. Think of a pendulum swinging, water rippling across a pond, or sound vibrations traveling through air. All of these follow sine wave patterns. Even the alternating current powering your home oscillates as a sine wave. The remarkable property that any waveform, no matter how complex, can be broken down into sine wave components makes this function indispensable. From the music you listen to, to the Wi-Fi signals carrying digital data, to ocean waves, sine waves underlie it all.

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