Trigonometry

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. /ˌsoʊkəˈtoʊə/ SOH-kə-TOH-ə, similar to Krakatoa).[33] Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".[34]

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.[35] 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 ${\displaystyle x=\cos A}$ and ${\displaystyle y=\sin A}$.[35] This representation allows for the calculation of commonly found trigonometric values, such as those in the following table:[36]

Function	0	${\displaystyle \pi /6}$	${\displaystyle \pi /4}$	${\displaystyle \pi /3}$	${\displaystyle \pi /2}$	${\displaystyle 2\pi /3}$	${\displaystyle 3\pi /4}$	${\displaystyle 5\pi /6}$	${\displaystyle \pi }$
sine	${\displaystyle 0}$	${\displaystyle 1/2}$	${\displaystyle {\sqrt {2}}/2}$	${\displaystyle {\sqrt {3}}/2}$	${\displaystyle 1}$	${\displaystyle {\sqrt {3}}/2}$	${\displaystyle {\sqrt {2}}/2}$	${\displaystyle 1/2}$	${\displaystyle 0}$
cosine	${\displaystyle 1}$	${\displaystyle {\sqrt {3}}/2}$	${\displaystyle {\sqrt {2}}/2}$	${\displaystyle 1/2}$	${\displaystyle 0}$	${\displaystyle -1/2}$	${\displaystyle -{\sqrt {2}}/2}$	${\displaystyle -{\sqrt {3}}/2}$	${\displaystyle -1}$
tangent	${\displaystyle 0}$	${\displaystyle {\sqrt {3}}/3}$	${\displaystyle 1}$	${\displaystyle {\sqrt {3}}}$	undefined	${\displaystyle -{\sqrt {3}}}$	${\displaystyle -1}$	${\displaystyle -{\sqrt {3}}/3}$	${\displaystyle 0}$
secant	${\displaystyle 1}$	${\displaystyle 2{\sqrt {3}}/3}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle 2}$	undefined	${\displaystyle -2}$	${\displaystyle -{\sqrt {2}}}$	${\displaystyle -2{\sqrt {3}}/3}$	${\displaystyle -1}$
cosecant	undefined	${\displaystyle 2}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle 2{\sqrt {3}}/3}$	${\displaystyle 1}$	${\displaystyle 2{\sqrt {3}}/3}$	${\displaystyle {\sqrt {2}}}$	${\displaystyle 2}$	undefined
cotangent	undefined	${\displaystyle {\sqrt {3}}}$	${\displaystyle 1}$	${\displaystyle {\sqrt {3}}/3}$	${\displaystyle 0}$	${\displaystyle -{\sqrt {3}}/3}$	${\displaystyle -1}$	${\displaystyle -{\sqrt {3}}}$	undefined

The following table summarizes the properties of the graphs of the six main trigonometric functions:[38][39]

Function	Period	Domain	Range	Graph
sine	${\displaystyle 2\pi }$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle [-1,1]}$
cosine	${\displaystyle 2\pi }$	${\displaystyle (-\infty ,\infty )}$	${\displaystyle [-1,1]}$
tangent	${\displaystyle \pi }$	${\displaystyle x\neq \pi /2+n\pi }$	${\displaystyle (-\infty ,\infty )}$
secant	${\displaystyle 2\pi }$	${\displaystyle x\neq \pi /2+n\pi }$	${\displaystyle (-\infty ,-1]\cup [1,\infty )}$
cosecant	${\displaystyle 2\pi }$	${\displaystyle x\neq n\pi }$	${\displaystyle (-\infty ,-1]\cup [1,\infty )}$
cotangent	${\displaystyle \pi }$	${\displaystyle x\neq n\pi }$	${\displaystyle (-\infty ,\infty )}$

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.[40]^: 48ff

The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:[40]^: 48ff[41]^: 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:[42]

${\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\\end{aligned}}}$

${\displaystyle {\begin{aligned}\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.\end{aligned}}}$

With these definitions the trigonometric functions can be defined for complex numbers.[43] When extended as functions of real or complex variables, the following formula holds for the complex exponential:

${\displaystyle e^{x+iy}=e^{x}(\cos y+i\sin y).}$

This complex exponential function, written in terms of trigonometric functions, is particularly useful.[44][45]

Trigonometric functions were among the earliest uses for mathematical tables.[46] 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.[47] Slide rules had special scales for trigonometric functions.[48]

Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).[49] 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.[50] The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.[51]

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[52]), the coversine (coversin(θ) = 1 − sin(θ) = versin(π/2 − θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin²(θ/2)),[53] 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.

For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,[54] predicting eclipses, and describing the orbits of the planets.[55]

In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,[56] as well as in satellite navigation systems.[16]

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.[57]

Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.[58]

In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.[59]

On a larger scale, trigonometry is used in geography to measure distances between landmarks.[60]

Function ${\displaystyle s(x)}$ (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, ${\displaystyle S(f)}$ (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).

The sine and cosine functions are fundamental to the theory of periodic functions,[61] such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.

Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics[62] and communications,[63] among other fields.

Trigonometry is useful in many physical sciences,[64] including acoustics,[65] and optics.[65] In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.[66]

Other fields that use trigonometry or trigonometric functions include music theory,[67] geodesy, audio synthesis,[68] architecture,[69] electronics,[67] biology,[70] medical imaging (CT scans and ultrasound),[71] chemistry,[72] number theory (and hence cryptology),[73] seismology,[65] meteorology,[74] oceanography,[75] image compression,[76] phonetics,[77] economics,[78] electrical engineering, mechanical engineering, civil engineering,[67] computer graphics,[79] cartography,[67] crystallography[80] and game development.[79]

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).[83]

The law of sines (also known as the "sine rule") for an arbitrary triangle states:[84]

${\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R={\frac {abc}{2\Delta }},}$

where ${\displaystyle \Delta }$ is the area of the triangle and R is the radius of the circumscribed circle of the triangle:

${\displaystyle R={\frac {abc}{\sqrt {(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}}.}$

The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:[84]

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}$

or equivalently:

${\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}$

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.[85] It is given by:

${\displaystyle {\frac {a-b}{a+b}}={\frac {\tan \left[{\tfrac {1}{2}}(A-B)\right]}{\tan \left[{\tfrac {1}{2}}(A+B)\right]}}}$

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:[84]

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

${\displaystyle s={\frac {1}{2}}(a+b+c),}$

then the area of the triangle is:[86]

${\displaystyle {\mbox{Area}}=\Delta ={\sqrt {s(s-a)(s-b)(s-c)}}={\frac {abc}{4R}}}$,

where R is the radius of the circumcircle of the triangle.

${\displaystyle {\mbox{Area}}=\Delta ={\frac {1}{2}}ab\sin C.}$

The following trigonometric identities are related to the Pythagorean theorem and hold for any value:[87]

${\displaystyle \sin ^{2}A+\cos ^{2}A=1\ }$

${\displaystyle \tan ^{2}A+1=\sec ^{2}A\ }$

${\displaystyle \cot ^{2}A+1=\csc ^{2}A\ }$

The second and third equations are derived from dividing the first equation by ${\displaystyle \cos ^{2}{A}}$ and ${\displaystyle \sin ^{2}{A}}$, respectively.

Euler's formula, which states that ${\displaystyle e^{ix}=\cos x+i\sin x}$, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:

${\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}},\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}},\qquad \tan x={\frac {i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}}.}$

Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.[29]