Mathematics: Trigonometry

Introduction

Trigonometry is the branch of mathematics that deals with the properties of triangles and of the trigonometric functions, which were originally developed to describe certain properties of triangles but have applications in math and the sciences far beyond flat geometry.

For about 2,000 years, trigonometric functions were especially prominent in astronomy and geography, where they were first developed. Today they are part of the daily working language of scientists and engineers in a multitude of fields.

For centuries after their discovery, the trigonometric functions—of which the most familiar and fundamental are those known as the sine and cosine—were closely associated with the properties of triangles. Only in the 1600s did mathematicians free them from direct association with triangles and bring them into the family of general functions on a real axis. First, the Newton-Taylor expansion of smooth functions as infinite sums (series) of polynomials connected the trigonometric functions to algebraic functions. In the eighteenth century Leonard Paul Euler (1707–1783) connected them to exponential functions using complex numbers. In the nineteenth century, French mathematician Joseph Fourier (1768–1830) developed his analysis, which became a powerful tool for analyzing signals in scientific and technological settings. Today trigonometric functions find application throughout much of higher and applied mathematics.

Historical Background and Scientific Foundations

The word “trigonometry” is from the Greek for “the geometry of triangles.” However, the first applications of trigonometry probably did not originate with the Greeks—there are indications that Greeks learned some trigonometric ideas from the Egyptians and Babylonians. For example, it is possible that the Greek thinkers Pythagoras (c.582–c.500 BC) and Plato (c.427–c.347 BC) learned mathematics in Egypt. Alexander the Great's (356–323 BC) military exploits in 333–323 BC briefly united the Mediterranean and Middle East under Greek rule, and sharing of scientific ideas between different regions was more likely under these conditions. It was at about this time that the Greeks elevated science and mathematics to new levels of precision and abstraction. The Elements of Euclid (fl. third c. BC), written in the third century BC in Alexandria, Egypt, defined the field of geometry for the next 2,000 years.

Euclid developed the idea of similarity between triangles of different sizes but identical angles. This concept allowed measurement of the size of Earth using nothing but simple geometry and a vertical stick, a device called a gnomon (pronounced NO-mun). Using a gnomon, Eratosthenes (276–194 BC), measured the angular height of the sun in Alexandria on the day

of the summer solstice (point on the calendar when the days cease to get longer and begin to get shorter again). Knowing that in another Egyptian city, Thebes, the sun stood directly overhead on the same day at the same hour, Eratosthenes could compute the angular difference between the two cities on the surface of the Earth, which Greek thinkers understood correctly to be spherical. (It is not true that Christopher Columbus [1451–1506] discovered, proved, or was the first to suggest that Earth is round.) Eratosthenes estimated that Earth's circumference is 50 times greater than the distance between Thebes and Alexandria—a reasonably correct value achieved using extremely simple tools.

Euclid did not introduce any of the specific functions associated with the angles, the trigonometric functions. (A function is a specific relationship between two or more sets of numbers.) Neither did the Greek scientist Archimedes (287–212 BC), who in his treatise Measurement of a Circle (c.240 BC) computed accurate bounds on p. In doing so, Archimedes computed estimates of the chords (circle-bridging line segments) associated with certain angles, namely those that are multiples of 3.75°. Today, his procedure would be viewed as the computation of the sines and tangents of those angles—two of the basic trigonometric functions.

In astronomy and physics, the Aristotelian idea that circular motion is more natural, perfect, and pervasive was dominant for over 1,500 years. Since the heavenly bodies clearly do not move in simple circles, the idea of epicycles, circles moving along other circles, evolved. Hipparchus (190–120 BC), an astronomer of Rhodes (a Greek island), was the first to construct a viable astronomical system, around 160–140 BC. This might have been the earliest use of trigonometry in astronomy, for although Hipparchus never thought to introduce a specific function analogous to the modern sine [sin(x), read aloud as “sine x”], in his system the position of the sun was expressed by a formula that implicitly included sines.

Other early astronomers pushed trigonometry to greater sophistication. The Almagest of Claudius Ptolemy (AD c.90–c.168), composed around AD 150 in Alexandria, was the central textbook of ancient and medieval astronomers for about 1,400 years. It was supplanted only when Nicolaus Copernicus (1473–1543) wrote his De Revolutionibus (1543), which placed the sun, rather than Earth, at the center of the universe.

Despite the downfall of Rome and the advent of the Dark Ages in Europe, the Almagest was preserved by translation into Arabic (four different translations were made in ninth-century Baghdad). The first translation from Arabic back into Latin by Gerard of Cremona (c.1175) marked the scientific re-awakening of Europe.

In the Almagest, Ptolemy claimed to explain all of the phenomena in the sky, including the most intricate and puzzling then known, namely, the retrograde motions of the planets (that is, their apparent motion,

IN CONTEXT: THE TRIGONOMETRIC FUNCTIONS

Two basic mathematical relationships, the sine and cosine functions, arise in the study of triangles. Various simple ratios of these produce a family of secondary or derived functions, namely the tangent, cotangent, secant, cosecant. Together, these functions are known as the trigonometric functions.

Given a particular angle,.(Greek letter theta), each trigonometric function specifies a number. For example, the sine of 45°, written sin(45°), is the square root of 2 (≅.7071); the sine of 90° is 1; and so on.

The sine of an angle is found by assuming that the angle is inside a right triangle (a triangle having one right or 90° angle). The longest side of any right triangle is the side facing (across from) its right angle: this side is called the hypotenuse. The sine of any angle in a right triangle is the length of the side facing that angle divided by the length of the hypotenuse. The sine of a 90° angle, for example, is 1, because it is given by the length of the hypotenuse divided by itself.

The cosine of an angle is defined as the length of the side of the triangle adjacent to the angle divided by the length of the hypotenuse.

Both the sine and cosine, plotted in Cartesian coordinates with the angle ϑ on the horizontal axis and the magnitude of the function on the vertical axis, appear as wavy, up-and-down lines that repeat forever to left and right.

All other trigonometric functions are formed as simple ratios of the sine and cosine. The tangent is sine/cosine, the cotangent is cosine/sine, the secant is 1/cosine, and the cosecant is 1/sine. These functions are a useful shorthand but contain no information that is not found in the sine and cosine. Indeed, since sin(ϑ) = cos(90° -ϑ) and cos(ϑ) = sin(90° -ϑ), either of the sine or cosine alone would suffice to write down all other trigonometric functions. Such notation would, however, be obscure and inconvenient.

In modern mathematics, definitions of the sine and cosine that do not depend on the properties of triangles are sometimes used so that these functions can be extended to complex numbers.

at times, from east to west relative to the fixed stars, opposite to the west-to-east motions of all other heavenly bodies). The Almagest explicitly introduced the sine function, though under a different name. Ptolemy sought to use sine functions to characterize the error introduced in estimating the size of Earth using the gnomon by the fuzziness of the gnomon's shadow, which introduces angular uncertainty into observations of the sun's position.

Muslim mathematicians made further progress in trigonometry during the long sleep of European science. Their main goal was to find Kibla—the direction to Mecca from any locality on Earth—so that Muslims everywhere could face Mecca during prayer. In the eleventh century, Muslim scientists discovered a number of cosine and sine theorems. Later Muslim mathematics concentrated on producing more precise Kibla tables.

The introduction of the tangent function (as well as cotangent, secant, and cosecant) is attributed to Muslim mathematician Abu'l-Wafa (959–988), a member of the Caliph's court in Baghdad. Though these functions are simply different ratios of sine and cosine functions, their development allowed the use of a shorter, more useful algebraic notation that was important in the development of mathematics in medieval Europe.

The need of medieval European traders for convenient, accurate maps dictated further development of trigonometry. Since Earth is round, while maps on paper must be flat, mapmaking requires mathematical understanding of the projection of the surface of a sphere onto a plane. Around 1569, Flemish cartographer Gerardus Mercator (1512–1594) found a convenient way to represent part of the globe on a flat map using logarithms and trigonometric functions.

The Mercator map projection, with its straight meridians and parallels, was widely used for navigation and is still one of the most common projections. The projection has its advantages, but latitude (distance from the equator) is distorted, with distortion increasing with distance from the poles until it approaches infinity at the poles. A Mercator map thus shows more or less accurate proportions near the equator, but is less accurate near the North and South Poles. For example, on a Mercator projection Greenland looks as big as South America, though it is in fact only a tenth as large.

The Mercator projection was the first case in the history of applied mathematics where the logarithm (inverse of the exponential function) and trigonometric functions intermingled. Previously, they had seemed to have little relationship.

In the seventeenth century, progress in trigonometry continued to be driven by astronomy. German astronomer Johannes Kepler (1571–1630) was the first to suggest, in his 1609 Astronomia Nova, that all planets move in elliptical orbits, rather than in circles (or circles moving in circles). His basic equation to relate the position of the sun to that of any given planet (e.g., Earth) included a sine function.

IN CONTEXT:TRIGONOMETRY AND NAVIGATION

Pilots, mariners, and mountaineers all use trigonometric concepts to find their way from one point to another. In navigation, positive angles are measured clockwise from North and are known as azimuths. Azimuths convey direction, so they can range from 0° to 360°, and the word azimuth is sometimes used synonymously with the word heading. The azimuth of a line running from south to north is 0° and the azimuth of a line running from north to south is 180°. This distinction is critical in navigation. In other applications, it may not be critical to distinguish the direction. For example, it does not matter whether the boundary of a country runs from north to south or south to north.

In navigation involving travel over large distances, Earth's curvature becomes important and spherical rather than plane; trigonometry must be used. Coordinates for navigation over long distances are given in terms of latitude and longitude, which are angular measurements. Trigonometry is used to calculate the distance between the starting and ending points of a journey, taking into account that the path follows the surface of a sphere and not a straight line. The latitude and longitude of waypoints along a journey can also be calculated using trigonometry.

Navigation on Earth is complicated by the fact that the North Magnetic Pole, to which compass needle's are attracted, does not coincide with the North Geographic Pole. The North Magnetic Pole is located in far northern Canada. For very approximate navigation, for example if a hiker wants to know if she is generally headed north or south, the fact that the geographic and magnetic poles are different does not make much difference. For any kind of precise navigation or mapmaking, however, the difference is important. The difference between true north, which is the direction to the North Geographic Pole, and magnetic north, which is the direction to the North Magnetic Pole, is known as magnetic declination. It is shown as an angle on topographic maps and navigational charts.

Magnetic north is about 20° east of true north in the northwestern United States and about 20° west of true north in the northeastern United States. The line of zero declination runs through the midwestern part of the country. In other areas of the world, the magnetic declination can be as great as 90° east or west in the far southern hemisphere. The North Magnetic Pole moves from year to year as a consequence of Earth's rotation, so the magnetic declination also changes over time. Government agencies responsible for providing navigation aids track the movement of the North Magnetic Pole, and maps are continually revised to reflect changing declination. Measurements by the Canadian government show that the North Magnetic Pole moved an average of 25 mi (40 km) per year between 2001 and 2005.

A simple trigonometric calculation illustrates the error that can occur if magnetic declination is not taken into account. The distance off course will be the distance traveled multiplied by the sine of the magnetic declination. In an area where the magnetic declination is 20°, therefore, a sailor following a course due north would find herself 21.1 mi (34 km) off course at the end of a 62.1 mi (100 km) trip. The longer the distance traveled, the farther off course the traveler will be. If the magnetic declination is only 10°, however, the error will be 62.1 mi (100 km) × sin 10° = 10.6 mi (17 km).

The application of trigonometric functions to technological innovation also began in the seventeenth century. One of the interests of mid-seventeenth century mathematicians was to find the area under a curve; avorite curves were the cycloid—the curve described in the air by a point on a rim of a rolling circular wheel—and a more general curve, the trochoid (or prolate cycloid), described by a point located inside the wheel.

Consideration of the cycloid, which requires the use of trigonometric functions, gave the solution to the tau-tochrone problem. The tautochrone is a curve, which, if imagined as a stiff wire along which frictionless beads slide, allows all beads to reach the bottom of the curve at the same time regardless of their starting point. The Dutch astronomer Christiaan Huygens (1629–1695) proved in 1673 that the cycloid curve is the tautochrone curve. Huygens used this curve to construct the first pendulum clock and to describe methods for improving the accuracy of pendulum clocks at sea.

The relationship of trigonometric functions to polynomials and exponential functions was the next area of progress, and eventually had important implications for technology and science. Isaac Newton's disciple, English mathematician Brook Taylor (1684–1731), discovered a method of expanding any function in the vicinity of a point, later known as Taylor's Theorem. The resulting expansion is called a Taylor expansion. In mathematics, an expansion is an expression for a function that consists of a sum of terms, perhaps an infinite number of them. In particular, Taylor showed that trigonometric functions can be represented as infinite Taylor expansions: for example, sin(x) can be expressed as a sum of powers of x multiplied by various constants. Taylor series expansions enabled mathematicians to recognize the relationship between trigonometric functions and logarithms, which can also be expressed as Taylor series.

Developing the Newton-Leibniz calculus, Swiss mathematician Leonard Euler (1707–1783) discovered,

IN CONTEXT: VECTORS, FORCES, AND VELOCITIES

Vectors are quantities that have both direction and magnitude, for example the velocity of an automobile, airplane, or ship. The direction is the azimuth in which the vehicle is traveling and the magnitude is its speed. Using trigonometry, vectors can also be broken down into perpendicular components that can be added or subtracted. Take the example of a ferry that carries cars and trucks across a large river. If there are ferry docks directly across from each other on opposite banks of the river, the captain must steer the ferry upstream into the current in order to arrive at the other dock. Otherwise, the river current would push the ferry downstream and it would miss the dock. If the velocities of the river current and the ferry are known, then the captain can calculate the direction in which he must steer to end up at the other dock. The velocity of the river current forms one leg of a right triangle and the velocity of the ferry forms the hypotenuse (because the captain must point the ferry diagonally across the river to account for the current).

If the current is moving at 3.1 mph (5 km/hr) and the ferry can travel at 7.5 mph (12 km/hr), the angle at which the ferry needs to travel is found by calculating its sine. In this case, the sine of the unknown angle is 5/12 = 0.4167. The angle can then be determined by looking in a table of trignometric functions to find the angle that most closely matches the calculated value of 0.4167, by using a calculator to calculate the sines of different angles and comparing the results, or by using the arc sine (asin) function. Each of the trigonometric functions has an inverse function that allows the angle to be calculated from the value of the function. In this case, the answer is sin 0.4167 = 25°. In other words, the captain must point his ferry 25° upstream in order to account for the current and arrive at the dock directly across the river.

Another application of vectors and trigonometry involves weight and friction. Automobiles and trains rely on friction to move uphill or remain in place when parked, and friction is required in order to hold soil and rock in place on steep slopes. If there is not enough friction, cars will slide uncontrollably downhill, and landslides will occur. Even if a car is traveling downhill, friction is required to steer. In the simplest case, the traction of a vehicle or the resistance of a soil layer to landsliding depends on three things: the weight of the object, the coefficient of friction, and the steepness of the slope. The weight of the object is self-explanatory. The coefficient of friction is an experimentally measured value that depends on the two surfaces in contact with each other and, in some cases, temperature or the rate of movement. The value used before movement begins, for example between the tires of a parked car and the pavement or a soil layer that is in place, is known as the static coefficient of friction. Once the object begins moving, the coefficient of friction decreases and is known as the dynamic coefficient of friction. Some typical examples of coefficients of friction are 0.7 for tires on dry asphalt, 0.4 for tires on frosty roads, and about 0.2 for tires on ice. The coefficients of friction for soils involved in landslides can range from about 0.3 to 1.0, with most values around 0.6.

Because weight is a force that acts vertically downward, trigonometry must be used to calculate the components of weight that are acting parallel to the sloping surface. The frictional force resisting movement parallel to the slope is µ × w × cosϑ, where w is the weight, µ is the coefficient of friction, and ϑ is the slope angle. The component of the weight acting downslope is w × sinϑ. Division of the frictional resisting force by the gravitational driving force gives the expression µ / tanϑ. If the result is equal or greater than 1, the car or soil layer will not slide downhill. If it is less than 1, then downhill sliding is inevitable. If the coefficient of friction for tires on dry asphalt is 0.7, then parked cars will slide downhill if the slope is greater than 35°. If the road is covered with ice, however, the coefficient of friction is only 0.2 and cars will slide downhill on slopes greater than 11°.

using Taylor series, that a complex exponential can be expressed as a sum of two trigonometric functions, one multiplied by the square root of -1. This discovery opened the way to many results in mathematics and technology. In particular, the use of Euler's identity allows the handling of periodic phenomena—events that repeat regularly in time, like the voltage of ordinary alternating-current power—using complex exponentials, which, despite their formidable-sounding name, are easier to manipulate mathematically than sinusoids. Today, complex exponentials are a standard tool in such fields as electrical engineering, where sinusoidal signals are commonplace.

The last major innovations in the fundamental theory of trigonometric functions came in the nineteenth century; since that time, there has been little mathematical innovation in this area. In the late nineteenth century, the development of non-Euclidean geometries—which would be essential to the development of general relativity theory in the early twentieth century—triggered the exploration of non-Euclidean trigonometries, in which, for example, the sum of the interior angles of a triangle is not 180°, as in ordinary plane geometry. In 1822, Fourier discovered that any periodic function (one that repeats itself exactly over a fixed time period), no matter how complex its shape or even if it contains sudden jumps or discontinuities, can be written as a superposition or adding-up of sine functions shifted and amplified by different amounts. That is, any periodic waveform or function can be built by piling up appropriately shifted and scaled sinusoids. A sum of sinusoids representing a particular periodic function is called a Fourier series.

Modern Cultural Connections

Trigonometric functions are well-established mathematical tools that permeate the working equations of mathematics, social and physical science, and technology. There is no controversy about their basic nature or validity.

Fourier's insight—that one realm of functions can be mapped back and forth to another—extended in the nineteenth century to non-periodic signals (in the Fourier transform) and in twentieth century to discrete functions consisting of lists of numbers (in the discrete Fourier transform) and has found application throughout science and technology. It is by using such transform techniques—rooted directly in trigonometric functions and their close relatives, complex exponentials—that scientists can examine the frequency spectra of data gathered over time, such as audio signals or geological markers of ancient climate variations. Digital filtering of sampled audio signals often relies on the discrete Fourier transform and related techniques.

Significant developments in the application of sinusoidal functions have occurred at least as recently as the 1980s, when modern wavelet transforms were developed. These are now commonly applied in signal processing (for example, in removing noise from audio and video recordings). In some applications, wavelet techniques are superior to Fourier analysis, for example, chemists rely on trigonometry to analyze unknown substances using methods such as Fourier transform spectroscopy.

Trigonometric principles and calculations have applications in virtually every discipline of science and engineering, so their importance will continue to increase as technology continues to grow in importance, especially in fields such as computer and global positioning system technologies.

Computer Technologies

Both two-and three-dimensional computer graphics applications make heavy use of trigonometric relationships and formulae. Rotating an object in two dimensions, for example a spinning object in a video game or the text in an illustration, requires calculation of the sine and cosine of the angle through which the object is being rotated. Graphics objects are typically defined using points for which x and y coordinates are known. In some cases, the points may represent the ends of lines or the vertices of polygons. Many computer programs that allow users to rotate objects require the user to enter an angle of rotation or use a graphics tool that allows for freehand rotation in real time. Each time a new angle is entered or the mouse is moved to rotate an object on the screen, the new coordinates of each point must be quickly calculated.

Rotation of graphical objects in three dimensions is much more complicated than it is in two dimensions. This is because instead of one angle of rotation, three angles must be given. Although there are several different conventions that can be used to specify the three angles of rotation, the one that is most understandable to many people is based on roll, pitch, and yaw. These terms were originally used to describe the three kinds of rotation experienced in a ship as it moves across the sea and were adopted to describe the motion of aircraft in the twentieth century. Roll refers to the side-to-side rotation of a ship or aircraft around horizontal axis. An aircraft is rolling if one of its wings is going up as the other goes down. Pitch refers to the upward or downward rotation of the bow of a ship or the nose of an aircraft. As the bow or nose goes up, the stern or tail goes down and vice versa. The final component of three-dimensional rotation is yaw, which refers to the side-to-side rotation of the nose or bow around a vertical axis. Just as in two-dimensional graphics, the simulation of three-dimensional rotation by a computer program requires that trigonometric functions be calculated for each of the three angles and applied to each point or polygon vertex. Three dimensional graphics are also more complicated because the shape of each object being simulated must be projected onto a two dimensional computer monitor or other plane, just as Earth's spherical surface must be projected to make a map.

GPS Technology

Global positioning system (GPS) receivers embedded in cellular telephones, vehicles, and emergency transmitters will allow lost travelers to be located and criminal suspects to be tracked. Fast Fourier transforms will help to advance any kind of computation involving waveforms, including voice recognition technologies. Trigonometric calculations related to navigation will also become even more important as global air travel increases and the responsibility for air traffic control is increasingly shifted from humans to computers and GPS technology.

Even when GPS receivers are used to determine the locations of unknown points, the locations of known points are used to increase accuracy. This is done by placing one GPS receiver over a known point and using a second receiver at the point for which a location must be determined. In the United States, one of the known points might be a continuously operating reference station, or CORS, operated by the government and providing data to surveyors over the Internet. Data from the two receivers are combined, either in real time or afterwards by post-processing, to obtain a more accurate solution that can be accurate to a millimeter or so. Although it may not be obvious because the calculations are performed by microprocessors within the GPS receivers and on computers, they require extensive use of trigonometric functions and principles.

Once the locations of points or features are determined, they must be plotted on a map in order to be visualized. If the area of concern is relatively small, the map can be constructed using an orthogonal grid system of perpendicular lines measuring the north-south and east-west distance from an arbitrary point. If the area to be mapped is large, however, then trigonometry must be used to project the nearly spherical surface of Earth onto a flat plane. Over the centuries, cartographers and mathematicians have developed many specialized projections involving trigonometric functions. Some are designed so that angles measured on the flat map are identical to those measured on a round globe, some are designed so that straight-line paths on the globe are preserved as straight lines on the planar map.

See Also Earth Science: Geodesy.

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Monmonier, Mark. Rhumb Lines and Map Wars: A Social History of the Mercator Projection. Chicago: University of Chicago Press, 2004.

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Duke, D. “The Equant in India: The Mathematical Basis of Indian Planetary Models.” Archive for History of Exact Sciences 59 (2005): 563–576.

Ari Belenkiy