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Trigonometry
In geometry, study of the relationship between angles and lengths
Trigonometry is known for its many identities,^{[5]}^{[6]} which are equations used for rewriting trigonometrical expressions to solve equations, to find a more useful expression, or to discover new relationships.^{[7]}
History
Hipparchus, credited with compiling the first trigonometric table, has been described as "the father of trigonometry".^{[8]}
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.^{[9]} They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.^{[10]}
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.^{[11]} In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.^{[12]} Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.^{[13]} (The value we call sin(?) can be found by looking up the chord length for twice the angle of interest (2?) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.
The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.^{[14]} These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.^{[15]}^{[16]} The PersianpolymathNasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.^{[17]}^{[18]}^{[19]}Nas?r al-D?n al-T?s? was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.^{[20]} He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.^{[21]} Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.^{[22]} One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years.^{[23]} At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.^{[24]} Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.^{[25]}Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.^{[26]}Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.^{[27]} Also in the 18th century, Brook Taylor defined the general Taylor series.^{[28]}
Trigonometric ratios
In this right triangle: sin A = a/c;cos A = b/c;tan A = a/b.
Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under Mnemonics.
Since any two right triangles with the same acute angle A are similar^{[29]}, the value of a trigonometric ratio depends only on the angle A.
The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".^{[30]}
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.^{[31]} These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
Mnemonics
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:^{[32]}
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-ka-toe-uh' ). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".^{[33]}
The unit circle and common trigonometric values
Fig. 1a - Sine and cosine of an angle ? defined using the unit circle.
Trigonometric ratios can also be represented using the unit circle, which is the circle of radius 1 centered at the origin in the plane.^{[34]} In this setting, the terminal side of an angle A placed in standard position will intersect the unit circle in a point (x,y), where $x=\cos A$ and $y=\sin A$.^{[34]} This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:^{[35]}
Function
0
$\pi /6$
$\pi /4$
$\pi /3$
$\pi /2$
$2\pi /3$
$3\pi /4$
$5\pi /6$
$\pi$
sine
0
$1/2$
${\sqrt {2}}/2$
${\sqrt {3}}/2$
1
${\sqrt {3}}/2$
${\sqrt {2}}/2$
$1/2$
0
cosine
1
${\sqrt {3}}/2$
${\sqrt {2}}/2$
$1/2$
0
$-1/2$
$-{\sqrt {2}}/2$
$-{\sqrt {3}}/2$
-1
tangent
0
${\sqrt {3}}/3$
$1$
${\sqrt {3}}$
undefined
$-{\sqrt {3}}$
$-1$
$-{\sqrt {3}}/3$
0
secant
1
$2{\sqrt {3}}/3$
${\sqrt {2}}$
$2$
undefined
$-2$
$-{\sqrt {2}}$
$-2{\sqrt {3}}/3$
-1
cosecant
undefined
$2$
${\sqrt {2}}$
$2{\sqrt {3}}/3$
1
$2{\sqrt {3}}/3$
${\sqrt {2}}$
$2$
undefined
cotangent
undefined
${\sqrt {3}}$
$1$
${\sqrt {3}}/3$
0
$-{\sqrt {3}}/3$
$-1$
$-{\sqrt {3}}$
undefined
Trigonometric functions of real or complex variables
Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments^{[36]} (see trigonometric function).
Graphs of trigonometric functions
The following table summarizes the properties of the graphs of the six main trigonometric functions:^{[37]}^{[38]}
Function
Period
Domain
Range
Graph
sine
$2\pi$
$(-\infty ,\infty )$
$[-1,1]$
cosine
$2\pi$
$(-\infty ,\infty )$
$[-1,1]$
tangent
$\pi$
$x\neq \pi /2+n\pi$
$(-\infty ,\infty )$
secant
$2\pi$
$x\neq \pi /2+n\pi$
$(-\infty ,-1]\cup [1,\infty )$
cosecant
$2\pi$
$x\neq n\pi$
$(-\infty ,-1]\cup [1,\infty )$
cotangent
$\pi$
$x\neq n\pi$
$(-\infty ,\infty )$
Inverse trigonometric functions
Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible.^{[39]}^{:48ff}
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:^{[39]}^{:48ff}^{[40]}^{:521ff}
When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations:^{[41]}
With these definitions the trigonometric functions can be defined for complex numbers.^{[42]} When extended as functions of real or complex variables, the following formula holds for the complex exponential:
$e^{x+iy}=e^{x}(\cos y+i\sin y).$
This complex exponential function, written in terms of trigonometric functions, is particularly useful.^{[43]}^{[44]}
Calculating trigonometric functions
Trigonometric functions were among the earliest uses for mathematical tables.^{[45]} Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy.^{[46]}Slide rules had special scales for trigonometric functions.^{[47]}
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).^{[48]} Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions.^{[49]} The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.^{[50]}
Other Trigonometric Functions
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (crd(?) = 2 sin), the versine (versin(?) = 1 - cos(?) = 2 sin^{2}) (which appeared in the earliest tables^{[51]}), the coversine (coversin(?) = 1 - sin(?) = versin( - ?)), the haversine (haversin(?) = versin(?) = sin^{2}),^{[52]} the exsecant (exsec(?) = sec(?) - 1), and the excosecant (excsc(?) = exsec( - ?) = csc(?) - 1). See List of trigonometric identities for more relations between these functions.
Applications
Astronomy
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,^{[53]} predicting eclipses, and describing the orbits of the planets.^{[54]}
Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.^{[56]}
In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.^{[58]}
On a larger scale, trigonometry is used in geography to measure distances between landmarks,^{[59]}
Periodic functions
Function $s(x)$ (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, $S(f)$ (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).
Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics^{[61]} and communications^{[62]}, among other fields.
Optics and Acoustics
Trigonometry is useful in many physical sciences,^{[63]} including acoustics,^{[64]} and optics^{[64]}. In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.^{[65]}
Triangle with sides a,b,c and respectively opposite angles A,B,C
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.^{[80]}
Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,^{[81]} relate both the sides and angles of a given triangle.
Triangle identities
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).^{[82]}
Law of sines
The law of sines (also known as the "sine rule") for an arbitrary triangle states:^{[83]}
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:^{[83]}
$c^{2}=a^{2}+b^{2}-2ab\cos C,$
or equivalently:
$\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$
Law of tangents
The law of tangents, developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables.^{[84]} It is given by:
Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:^{[83]}
Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
Euler's formula, which states that $e^{ix}=\cos x+i\sin x$, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary uniti:
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.^{[29]}
^"Al-Tusi_Nasir biography". MacTutor History of Mathematics archive. Retrieved . One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
^electricpulp.com. "?USI, NA?IR-AL-DIN i. Biography - Encyclopaedia Iranica". www.iranicaonline.org. Retrieved . His major contribution in mathematics (Nasr, 1996, pp. 208-214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
^Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN978-0-691-11485-9.
^A sentence more appropriate for high schools is "'Some Old Horse Came A''Hopping Through Our Alley". Foster, Jonathan K. (2008). Memory: A Very Short Introduction. Oxford. p. 128. ISBN978-0-19-280675-8.