Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. 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.
In the 3rd century BC,
Hellenistic mathematicians such as
Archimedes studied the properties of
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,
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. 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. Ptolemy used
chord length to define his trigonometric functions, a minor difference from the
sine convention we use today. (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) 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
Islamic, and, later, Western European worlds.
In this right triangle: sin A = a/h;cos A = b/h;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 h 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
Cosine function (cos), defined as the ratio of the
adjacent leg (the side of the triangle joining the angle to the right angle) to the hypotenuse.
Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
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
Since any two right triangles with the same acute angle A are
similar, the value of a trigonometric ratio depends only on the angle A.
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-".
With these functions, one can answer virtually all questions about arbitrary triangles by using the
law of sines and the
law of cosines. 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.
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:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
One way to remember the letters is to sound them out phonetically (i.e. /ˌsoʊkəˈtoʊə/SOH-kə-TOH-ə, similar to
Krakatoa). Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".
Fig. 1a – Sine and cosine of an angle θ defined using the unit circle
Indication of the sign and amount of key angles according to rotation direction
Trigonometric ratios can also be represented using the
unit circle, which is the circle of radius 1 centered at the origin in the plane. 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 and . This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:
Trigonometric functions of real or complex variables
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.: 48ff
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:: 48ff : 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:
With these definitions the trigonometric functions can be defined for
complex numbers. When extended as functions of real or complex variables, the following
formula holds for the complex exponential:
This complex exponential function, written in terms of trigonometric functions, is particularly useful.
Trigonometric functions were among the earliest uses for
mathematical tables. 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.Slide rules had special scales for trigonometric functions.
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes
cis and their inverses). 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. The
floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating 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(θ) = 2 sin(θ/2)), the
versine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables), the
coversine (coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)), the
haversine (haversin(θ) = 1/2versin(θ) = sin2(θ/2)), the
exsecant (exsec(θ) = sec(θ) − 1), and the
excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). See
List of trigonometric identities for more relations between these functions.
Function (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, (in blue), which depicts
frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
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. 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:
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
^"Nasir al-Din al-Tusi". MacTutor History of Mathematics archive. Retrieved 2021-01-08. 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.
^"ṬUSI, NAṢIR-AL-DIN i. Biography". Encyclopaedia Iranica. Retrieved 2018-08-05. 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.
^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.