Trigonometry

Summary

Trigonometry is the study of triangles, namely the relationship between their three angles and three sides. It is concerned with both planar (two-dimensional) and spherical (three-dimensional) triangles. Any situation that involves angles or triangles can be seen as an application of trigonometry. For example, surveyors often measure one distance and two angles of a triangle as aids in marking boundaries. Trigonometry provides the other two distances and the other angle. In fields such as optics or acoustics, light and sound waves can be expressed in terms of sine, one of the six trigonometric functions.

Definition and Basic Principles

Trigonometry is the study of triangles, namely the relationships that exist between the measures of the three interior angles and the lengths of the three sides. To solve a triangle, one must determine the measures of each of its angles and the lengths of each of its sides. This is frequently done by means of the trigonometric ratios, especially the two primary ones, sine and cosine.

In Euclidean geometry, the sum of the measures of the interior angles of any plane triangle is 180 degrees. (However, for spherical triangles that total is always more than 180 degrees and is not always the same.) Thus for any plane triangle, if two angle measures are known, the third is immediately deduced. By means of this fact and others, principally the Pythagorean theorem, the law of sines, and the law of cosines, any triangle can be solved or proven not to exist as long as three of the six measurements of the triangle are known.

There are two exceptions. First, if only the three angle measures are known, the triangle cannot be solved, since similar triangles have the same angle measures but different side lengths. Second, if two sides and an angle are known, but the angle is not between the two known sides, this gives rise to the so-called ambiguous case. Two different triangles can satisfy the three known facts. In this event, both triangles are given as solutions, unless additional information rules out one of these possibilities.

Finally, the trigonometric relationships are commonly viewed not simply as ratios but as functions. They are then studied and used in calculus and higher mathematics, as any other function would be.

Background and History

The Egyptian Rhind papyrus dates to about 1650 BCE. It contains several problems regarding the slopes of pyramids. The solutions to these problems are equivalent to finding the cotangent ratios, although there is no evidence that the Egyptians ever thought in terms of angles. In the second century BCE, Hipparchus of Nicaea developed a table of chord lengths in a circle depending on the central angle. Bisecting the chord produced a triangle inside the circle; the chord values led directly to the sine ratio. Because his table was essentially the world's first table of trigonometric values, Hipparchus is often called the inventor of trigonometry.

The six trigonometric functions had all been studied by Arab scholars by the tenth century, most commonly in the service of astronomy. It was not until the fifteenth century that trigonometry was finally seen as an independent discipline within mathematics and not simply as a tool for solving astronomical problems. This came as a result of the publication of De triangulis omnimodis (wr. 1464, pb. 1533; On Triangles, 1967) by Viennese astronomer Regiomontanus (Johannes Müeller). The word “trigonometry” was first applied to the subject in 1595 by Bartholomäus Pitiscus. Soon after, trigonometry was subjected to new methods of analysis and took its modern form.

How It Works

Triangles can be constructed either on planes or on spheres. In addition, the six trigonometric functions are used apart from any direct reference to triangles.

Spherical Trigonometry. A spherical triangle has its vertices on the surface of a sphere. The total measure of its three interior angles is always greater than 180 degrees but less than 540 degrees; the exact amount depends on the particular triangle. The vertices of the triangle are connected by three great arcs, where a great arc is a portion of a great circle, the circle whose circumference is the largest possible on that sphere. Like the line segment, its counterpart in two dimensions, a great arc is the shortest distance between two points on a sphere.

In the system of radian measure, lengths of arcs correspond directly to the measures of angles. Thus, in a spherical triangle whose angles are represented by A, B, and C, and whose corresponding opposite sides are represented by a, b, and c, one may din relationships among the angles and sides by thinking always in terms of angles. For example, (sin a)/(sin A) = (sin b)/(sin B) = (sin c)/(sin C), and cos a = (cos b)(cos c)+(sin b)(sin c)(cos A). These equations (or related forms of the cosine rule) can be used to solve spherical triangles.

Plane Trigonometry. A plane triangle is two-dimensional. Its three vertices are connected by line segments and the sum of the measures of its three interior angles is 180 degrees.

The six trigonometric ratios apply to right triangles, that is, triangles containing a 90-degree angle. In this case, the famous Pythagorean theorem also applies, namely that the sum of the squares of the two legs of the triangle equals the square of the hypotenuse. If any two sides or any two angles are known, the third can then be deduced. If all sides, or all angles and one side, are known, the trigonometric ratios are used to find any unknown values.

By extension, these ratios can be used to analyze any triangle, because a perpendicular line segment can be dropped from any vertex to a base of the triangle, thus introducing one or two right triangles into the original diagram. These oblique triangles are usually solved by means of two identities, or laws. The law of sines states that (sin A)/a = (sin B)/b = (sin C)/c, where the notation is the same as in the case of spherical triangles. The law of cosines asserts that c² = a²+b²−2ab(cos C). Similar forms of this law involve the cosine of angles A and B.

Trigonometric Functions. By the seventeenth century, the trigonometric ratios were considered as functions whose input was an angle expressed in radian measure. As functions, they can be graphed. All six graphs are periodic, meaning that they repeat themselves indefinitely. The sine and cosine functions have the so-called sine wave or sinusoid for their graphs. The other four functions have vertical asymptotes that divide the graphs into infinitely many pieces.

In calculus, derivative and antiderivative relationships among the trigonometric functions are established. Of primary importance are the ones involving sine and cosine: cosine is the derivative function of sine, while the negative sine is the derivative function of cosine.

Applications and Products

Navigation. Historically, navigators would measure angles to landmarks on Earth or in the sky and consult tables with known positions or distances to evaluate the position of the vessel. Beginning in the mid-twentieth century, the technology that formed the basis of the Global Positioning System (GPS) was developed. The United States deploys between twenty-four and thirty-two satellites that continually broadcast signals to Earth, including the precise time at which the signal was sent. A position on Earth can be thought of as a function of four variables: latitude, longitude, altitude, and time. If signals are received from four different satellites, the receiver is programmed to solve a system of equations and return the value of the four variables. The solution often involves the method of trilateration, in which a coordinate system is rotated through two different angles and the new coordinates are expressed in terms of sines and cosines of the angles.

Trigonometry is also useful when a navigator is attempting to steer a vessel on the sea or in the air in the presence of wind or sea currents. First, the navigator begins with the desired path, both its angular direction and its velocity. This becomes one side of a triangle. At the end of that side, the navigator attaches a second side representing the current, according to its direction. The lengths of these sides represent the velocities. Connecting the ends of these segments with a third side produces a triangle in which two side lengths and one angle are known. The length of the third side represents the velocity of the vessel after the effect of the current has been accounted for, a value called the ground speed. Its angular position determines the true course of the vessel. The ground speed is typically discovered by the law of cosines and the true course by the law of sines.

Surveying. Triangulation has been used for surveying since the sixteenth century. The surveyor marks two positions and precisely measures the distance between them. Angles are measured between each of these two positions and a third landmark. The surveyor now knows two angles of the triangle and the length of one of the sides. The remaining angle is then computed, while the lengths of the other two sides are determined by means of the law of sines. The surveyor uses one side of this triangle as a basis and repeats the process using a fourth landmark to complete a second triangle. By repeating this process, the surveyor is able to cover the region in question with a series of triangles, all of whose measurements are known. These measurements are typically used to determine property boundaries.

Harmonic Motion.Simple harmonic motion is one in which an object repeatedly cycles through the same set of positions. Examples (ignoring friction) include the vibration of a string, the motion of a pendulum, and the voltage of household electricity. All of these can be represented by the equation y = a sin (bt), where t represents the input (usually in units of time), a represents the amplitude of the motion, and b is a factor derived from the frequency of the motion. Knowing this formula, specific types of harmonic motion can be attained by solving for the variables a and b.

Force. Forces are represented by vectors because they have both magnitude and direction. Trigonometry can be used to solve for forces by considering the lengths of the vectors to be the magnitude of the associated force and the angles to come from the direction in which the forces are being applied. One example is finding the tensions in two wires that are holding an object. The weight of the object is itself a force and can be represented as a vector whose direction is vertical (meaning, pointing toward the center of Earth). The typical solution procedure is to break a vector into its horizontal and vertical components, so that the vector is now the hypotenuse of a right triangle whose legs are these components. Because the total tension in the wires must equal the opposite of the weight vector (so that the object is actually suspended in the air), a set of equations is obtained in the horizontal and vertical directions. The sine and cosine functions are always involved in these solutions, the results of which are needed by the engineers, who must ensure that the structure they are building will support the weight as required.

Careers and Course Work

Trigonometry is a branch of elementary mathematics. As such, many jobs for which trigonometry is important will also require higher levels of mathematics or physics. An interested college student will go beyond trigonometry and major in mathematics, physics, or engineering. Many of these students will go on to earn a master's or doctoral degree before beginning their careers. Those who work as mathematicians take courses in calculus, differential equations, probability and statistics, linear and abstract algebra, analysis, and other courses. They will teach, work as researchers in pure or applied mathematics, or apply mathematics to solve problems in government or industry. Most physicists, in addition to taking many courses in mathematics, will also study fields, waves, heat, mechanics, quantum physics, and electricity and magnetism. Most physicists work in research or in developing new technologies.

Surveyors will typically take classes in drafting and mechanical drawing in addition to trigonometry. They must pass exams to become licensed. They may work as geodetic surveyors, working on relatively large areas on Earth's surface or perhaps as marine surveyors whose interest is a body of water. They also serve as cartographers and work in other fields.

Other professions making use of trigonometry include machinists, sheet metal workers, and forensic scientists. Most of these professions are projected to have good to above-average job growth, with the exception of machinists, where the number of jobs is expected to decline slowly.

Social Context and Future Prospects

The importance of trigonometry is that it lies at the foundation of the work of mathematics, engineering, and physics. For example, through their experimentation, physicists learn the principles at work in the universe, from the astronomical to the subatomic scales. One example of this is the high-profile experimentation being done using particle accelerators. Physicists hope to use these to recreate conditions that existed immediately following the big bang. They expect to gain insight into the forces and energies that first shaped the universe and continue to do so.

In itself, trigonometry is socially neutral. It does not generate difficult social or ethical dilemmas. For this reason, there are no controversies concerning trigonometry or its applications.

Bibliography

Klintberg, Bo. “Hipparchus' 3600'-Based Chord Table and Its Place in the History of Ancient Greek and Indian Trigonometry.” Indian Journal of History of Science 40, no. 2 (2005): 169–203.

Maor, Eli. Trigonometric Delights. Princeton, N.J.: Princeton University Press, 1998.

McKeague, Charles P., and Mark D. Turner. Trigonometry. Belmont, Calif.: Thomson Brooks/Cole, 2008.

Smythe, Danielle. "Examples of Trigonometry Being Used in Construction Jobs." Chron, 20 May 2021,work.chron.com/examples-trigonometry-being-used-construction-jobs-29944.html. Accessed 11 Jan. 2021.

Sultan, Alan. “CORDIC: How Hand Calculators Calculate.” The College Mathematics Journal 40, no. 2 (March, 2009): 87–92.

Van Brummelen, Glen. The Mathematics of Heaven and Earth: The Early History of Trigonometry. Princeton, N.J.: Princeton University Press, 2009.