Trigonometric Functions

Trigonometric functions are functions of an angle, most often the sine, cosine, and tangent, together with their reciprocals. They arose from the study of ratios in right triangles, which made them the core tool of trigonometry. Their modern meaning comes from the function viewpoint: defined on the unit circle, they take any real number as input and return a value, and they repeat with a fixed period. That periodicity explains their role in describing oscillation, rotation, and waves.

They matter because they connect geometry, calculus, and analysis. Their derivatives and integrals follow simple rules when angles are measured in radians, and their power series and their link to the exponential function through complex numbers reveal their deeper structure. They also underpin Fourier series, which express general periodic signals as sums of sines and cosines, a foundation of modern signal processing.

This entry first covers the definitions, radian measure, basic identities, graphs, and inverse functions. It then turns to analytic characterizations, calculus, and Fourier expansions. Historical development from ancient astronomy to early modern notation follows. The final sections cover applications in surveying, physics, engineering, and computing, as well as generalizations such as the hyperbolic functions.

Definitions in right triangles and on the unit circle

For an acute angle $\theta$ in a right triangle, the three sides are named by their position relative to $\theta$: the opposite side, the adjacent side, and the hypotenuse, which is the longest side and lies opposite the right angle. The three basic ratios are $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}.$$ Students often remember these as SOH-CAH-TOA. Each ratio depends only on the angle, not on the size of the triangle, because triangles with the same angles have proportional sides. The reciprocal functions are $\csc\theta = 1/\sin\theta$, $\sec\theta = 1/\cos\theta$, and $\cot\theta = 1/\tan\theta$, and $\tan\theta = \sin\theta/\cos\theta$.

The right-triangle definition covers only angles between 0 and $\pi/2$, so the functions are extended with the unit circle, the circle of radius 1 centred at the origin of the coordinate plane. Place the vertex of the angle at the origin, with its initial side along the positive $x$-axis, and let the terminal side meet the circle at a point $P = (x, y)$. Then $\cos\theta = x$ and $\sin\theta = y$, and $\tan\theta = y/x$ wherever $x \neq 0$. For acute angles this agrees with the triangle definition, because the hypotenuse of that triangle is 1. Positive angles are measured counterclockwise and negative angles clockwise, and a rotation by a full turn returns the point to where it started, so the definitions apply to every real angle. The remaining functions are $\sec\theta = 1/x$, $\csc\theta = 1/y$, and $\cot\theta = x/y$, each undefined where its denominator vanishes.

The sign of each function depends on the quadrant containing the terminal side:

• Quadrant I (0 to $\pi/2$): all six functions are positive.
• Quadrant II ($\pi/2$ to $\pi$): sine is positive; cosine and tangent are negative.
• Quadrant III ($\pi$ to $3\pi/2$): tangent is positive; sine and cosine are negative.
• Quadrant IV ($3\pi/2$ to $2\pi$): cosine is positive; sine and tangent are negative.

The phrase "All Students Take Calculus" lists the quadrants in this order and gives the positive functions in each.

Standard values for the common angles follow from elementary geometry. For $\pi/4$, an isosceles right triangle with legs of length 1 has hypotenuse $\sqrt{2}$. For $\pi/6$ and $\pi/3$, an equilateral triangle with sides of length 2 is split in half to give a right triangle with sides 1, 2, and $\sqrt{3}$. The results are summarised below.

Notation carries some conventions that are easy to misread. The function name is written in roman type and the argument usually appears without brackets, as in $\sin\theta$. The expression $\sin^2\theta$ means $(\sin\theta)^2$, not the sine of $\theta^2$. A superscript $-1$ denotes an inverse function ($\sin^{-1}x$ or $\arcsin x$), not a reciprocal; the reciprocal is written $\csc x$. Degree symbols are added explicitly: $\sin 30^\circ$ means the sine of thirty degrees, which equals $\sin(\pi/6)$ once the angle is expressed in radians (the conversion is discussed in the next section).

The same symbols serve both angles and real numbers because an angle measured in radians is identified with a real number: the arc length it cuts from the unit circle. Under this identification, $\sin x$ is a function defined on the real numbers and not only on geometric angles. This identification is what allows the functions to be differentiated, integrated, and expanded in series, and it is why the rest of the article treats them as functions of a real variable.

Angle measure and the radian

An angle can be measured in several units, and the choice affects both arithmetic and the form of the formulas that use it. The degree is the older and more familiar unit. Its 360-part division of the full turn has no geometric necessity. The division of the circle into 360 degrees can be traced to the Babylonians, who used a sexagesimal system of numbers. The 360 degrees may have been related to a year of 360 days. The number is convenient because it has many divisors, so a circle splits evenly into halves, thirds, quarters, sixths and so on. Greek astronomers such as Hipparchus and Ptolemy used Babylonian astronomical data as the foundation for their own work, and the 360-part circle passed into the Greek and later European tradition with them.

The radian is defined by the geometry of the circle itself. The radian measure of a central angle is the length of the arc it cuts off on a circle divided by that circle's radius. One radian is therefore the angle whose intercepted arc is exactly as long as the radius, which is about 57.3 degrees. Because the ratio of two lengths is dimensionless, the same angle gives the same radian value on every circle. Going once around a circle gives a circumference of 2πr, so the full turn is 2π radians, and hence 2π rad = 360°. Its name has a short history: the term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast. In 1874, after a consultation with James Thomson, Muir adopted radian. Earlier writers called the unit the circular measure of an angle. In the SI, the radian was first classed as a supplementary unit in 1960, and it is now treated as a dimensionless derived unit.

Converting between the two systems follows from the single relation π rad = 180°. An angle in degrees becomes radians by multiplying by π/180, and an angle in radians becomes degrees by multiplying by 180/π. Common values therefore read as 30° = π/6, 45° = π/4, 60° = π/3 and 90° = π/2. Scientific and engineering work usually adopts radians for quantities that involve rates of rotation. For example, radians per second is the standard unit for rotational speed, and angular velocity is expressed as the time derivative of the angle in radians. Degrees persist in fields such as navigation, surveying and observational astronomy, where a degree is further divided into 60 arcminutes and each arcminute into 60 arcseconds. In practice, the two units coexist, and a careful calculation states which one it uses.

Radians make many formulas simpler. The arc length of a circle is simply s = rθ when θ is in radians, with no conversion factor, while in degrees the same formula must carry a factor of π/180. The same economy appears in calculus. The derivative of sin x is cos x only when x is measured in radians; in degrees one obtains (π/180) cos x, and every later formula picks up a stray constant. The power series of sine and cosine, which are the basis for their analytic treatment through power series and Taylor series, likewise use radian measure. For the same reason, the approximation sin θ ≈ θ for small angles is valid only in radians. In mathematical writing, an angle given without a unit is conventionally taken to be in radians.

Angles also carry conventions of orientation. An angle is usually drawn in standard position, with its vertex at the origin and its initial side along the positive x-axis. A positive angle is measured counterclockwise, and a negative angle clockwise. Angles larger than a full turn or negative angles are both legitimate, and the function values repeat every 2π radians. These conventions differ from some practical systems. Compass bearings in navigation, for example, are measured clockwise from north, so a bearing of 90° points east rather than north. Common mistakes include confusing clockwise and counterclockwise directions or miscounting rotations. Keeping the orientation and the unit explicit is therefore part of working correctly with angles.

Radian | https://en.wikipedia.org/wiki/Radian Radian Measure and Arc Length | https://www.varsitytutors.com/practice/subjects/math-1/lessons/radian-measure-and-arc-length Radians & Arc Length — SL 3.3 | https://www.varsitytutors.com/practice/subjects/ib-mathematics-applications-and-interpretation/lessons/radians-and-arc-length History of measurement | https://en.wikipedia.org/wiki/History_of_measurement Ancient Greek use of Babylonian data (The Story Behind 360 Degrees) | https://www.thethinkacademy.com/blog/the-story-behind-360-degrees-why-circles-are-so-special/ A Complete Guide To Degrees In A Circle | https://completeera.com/a-complete-guide-to-degrees-in-a-circle-the-math-behind-360/ Earliest Known Uses of Some of the Words of Mathematics (R) | https://mathshistory.st-andrews.ac.uk/Miller/mathword/r/ graphsearch.epfl.ch (radian concept) | https://graphsearch.epfl.ch/en/concept/26003

Fundamental identities and symmetries

The most basic identity comes directly from the unit circle. A point at angle $\theta$ lies at $(\cos\theta, \sin\theta)$, and because every point of the circle is at distance 1 from the origin, the Pythagorean theorem gives $\sin^2\theta + \cos^2\theta = 1$ for every real $\theta$. Dividing this by $\cos^2\theta$ yields $1 + \tan^2\theta = \sec^2\theta$, and dividing by $\sin^2\theta$ yields $1 + \cot^2\theta = \csc^2\theta$. These variants hold wherever the functions involved are defined, that is, where the denominators are nonzero. Most simplifications of trigonometric expressions reduce to repeated use of these three forms.

Two symmetry properties describe how the functions behave under sign changes and shifts of the angle. Sine, tangent, cotangent, and cosecant are odd, meaning $\sin(-\theta) = -\sin\theta$, $\tan(-\theta) = -\tan\theta$, and so on, while cosine and secant are even, so $\cos(-\theta) = \cos\theta$. Geometrically, reflecting a point across the horizontal axis keeps its x-coordinate and flips its y-coordinate. The fundamental periods are $2\pi$ for sine, cosine, secant, and cosecant, and $\pi$ for tangent and cotangent. The shorter period of the tangent follows from the fact that $\sin(\theta+\pi) = -\sin\theta$ and $\cos(\theta+\pi) = -\cos\theta$: both change sign, so their quotient does not change. Sine itself is therefore not $\pi$-periodic, but it is anti-periodic with half-period $\pi$.

Co-function identities follow from the right triangle. The two acute angles of a right triangle add to $\pi/2$, so the sine of one equals the cosine of the other. This gives $\sin(\pi/2 - \theta) = \cos\theta$, $\cos(\pi/2 - \theta) = \sin\theta$, $\tan(\pi/2 - \theta) = \cot\theta$, and $\sec(\pi/2 - \theta) = \csc\theta$. The prefix "co-" refers to complementary angles. A related set of shift relations expresses one function as the other shifted by a quarter period: $\cos\theta = \sin(\theta + \pi/2)$ and $\sin\theta = \cos(\theta - \pi/2)$. These relations are the algebraic basis for the phase shifts discussed in the graphs section.

Addition and subtraction formulas

The sum formula for cosine can be derived from the distance between two points on the unit circle. Let $P = (\cos\alpha, \sin\alpha)$ and $Q = (\cos\beta, \sin\beta)$. Computing the squared distance directly gives $|PQ|^2 = 2 - 2(\cos\alpha\cos\beta + \sin\alpha\sin\beta)$. The same segment is the third side of an isosceles triangle with two sides of length 1 and included angle $\alpha - \beta$, so the law of cosines gives $|PQ|^2 = 2 - 2\cos(\alpha - \beta)$. Equating the two expressions yields

Replacing $\beta$ with $-\beta$ gives the formula for $\cos(\alpha+\beta)$, namely $\cos\alpha\cos\beta - \sin\alpha\sin\beta$. The sine formula follows from the co-function identity: $\sin(\alpha+\beta) = \cos\bigl((\pi/2 - \alpha) - \beta\bigr)$, and expanding this with the cosine difference formula gives $\sin\alpha\cos\beta + \cos\alpha\sin\beta$. Replacing $\beta$ with $-\beta$ gives $\sin(\alpha - \beta)$. The tangent sum formula is

which is undefined when $\tan\alpha\tan\beta = 1$ and is valid only where all the tangents involved exist.

Double-angle and half-angle formulas

Setting $\beta = \alpha$ in the addition formulas gives $\sin 2\alpha = 2\sin\alpha\cos\alpha$. The cosine double-angle formula has three equivalent forms: $\cos 2\alpha = \cos^2\alpha - \sin^2\alpha$, $\cos 2\alpha = 2\cos^2\alpha - 1$, and $\cos 2\alpha = 1 - 2\sin^2\alpha$. The last two are obtained from the first by applying the Pythagorean identity. Solving the forms $\cos 2\alpha = 1 - 2\sin^2\alpha$ and $\cos 2\alpha = 2\cos^2\alpha - 1$ for the squared functions with $\alpha = \theta/2$ gives the half-angle formulas

Taking square roots introduces a sign that is determined by the quadrant in which $\theta/2$ lies. The half-angle tangent has sign-free forms, $\tan\frac{\theta}{2} = \frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta}$, each valid where its denominator is nonzero. These identities turn squared or doubled arguments into single-angle expressions, which is why they are routinely used to simplify integrands and to evaluate products of trigonometric functions.

Graphs, periodicity, and transformations

The graphs of the six trigonometric functions follow directly from their definitions. Sine and cosine are smooth, bounded waves with range $[-1, 1]$. Each completes one cycle every $2\pi$ radians, and each has zeros spaced $\pi$ apart: sine crosses zero at integer multiples of $\pi$, while cosine crosses zero at odd multiples of $\pi/2$. Tangent, $\tan x = \sin x / \cos x$, has period $\pi$, vertical asymptotes at $x = \pi/2 + k\pi$, and zeros at $x = k\pi$. Its range is all real numbers. Cotangent has period $\pi$, asymptotes at $x = k\pi$, and zeros at odd multiples of $\pi/2$.

The reciprocal functions are built from these graphs. Cosecant, $\csc x = 1/\sin x$, has asymptotes wherever sine vanishes and no zeros; its values never lie strictly between $-1$ and $1$. Secant, $\sec x = 1/\cos x$, behaves the same way with the asymptotes shifted to $x = \pi/2 + k\pi$. Where a sine or cosine graph passes through $\pm 1$, the reciprocal graph touches the same points and turns back, forming the familiar U-shaped branches.

The general sinusoid

Most applied waves are written in the form $f(x) = A\sin(Bx + C) + D$. The four parameters act independently:

• Amplitude $|A|$ sets the distance from the midline to the peaks. A negative $A$ reflects the graph vertically but leaves the amplitude unchanged.
• Period $2\pi/|B|$ sets the length of one cycle. Larger $|B|$ compresses the graph horizontally.
• Phase shift is $-C/B$ when the argument is written as $B(x + C/B)$. Textbooks disagree on the sign convention. Some write $A\sin(B(x+C)) + D$, where a positive $C$ moves the graph left, while others write $\sin(Bx - C)$, where a positive $C/B$ moves it right. The safest practice is to rewrite the argument as $B(x - h)$ and read $h$ as the shift to the right.
• Vertical shift $D$ moves the midline to $y = D$, so the range becomes $[D - |A|, D + |A|]$.

These rules are not arbitrary. Multiplying the input by $B$ rescales the axis, adding a constant to the input translates it, and adding a constant to the output translates it vertically. Working through each step makes sign errors easier to catch.

Sine and cosine of the same frequency

Two sinusoids with the same frequency always combine into a single sinusoid. For any real constants $a$ and $b$,

where $\varphi$ is chosen so that $a = R\cos\varphi$ and $b = R\sin\varphi$. This is the phase form. It explains why a cosine wave is simply a sine wave shifted by a quarter period, and it is the basis for analysing alternating current, where voltage and current of equal frequency differ only by a phase angle (see alternating current).

Applied examples

Sinusoidal models are fitted from data by reading off the parameters. For a periodic quantity with maximum $M$ and minimum $m$, the amplitude is $A = (M - m)/2$ and the midline is $D = (M + m)/2$. The period is taken from the time between repeats, and the phase shift is chosen to match the time of a peak.

Tides provide a clean example. One source describes a harbour where the water depth varies between 2 m and 10 m with a 12-hour period, which gives $A = 4$, $D = 6$, and $B = \pi/6$ if $t$ is measured in hours from a high tide. The depth is then $d(t) = 6 + 4\cos(\pi t/6)$. Real tides are more complicated because they combine several lunar and solar constituents, and a single sinusoid is only a first approximation. Harmonic correction is used when more accuracy is needed.

Daylight hours follow the same pattern over a year. For a city with a minimum of 9.5 hours on day 355 and a maximum of 14.5 hours on day 172, the amplitude is 2.5 hours and the midline is 12 hours. Taking the period to be 365 days and placing the peak at day 172 gives $D(t) = 12 + 2.5\cos\bigl(2\pi(t - 172)/365\bigr)$. This form can be checked directly against the two extreme values, which is a useful habit when interpreting any fitted parameter.

Inverse trigonometric functions

Trigonometric functions are periodic, so each output value is produced by infinitely many input angles. An inverse therefore exists only after the domain has been restricted to an interval on which the function is one-to-one and still reaches every value in its range. The restricted function is called the principal branch, and the inverse built from it is the principal inverse.

Principal branches and their domains

The standard choices are as follows:

• Arcsine comes from the sine function restricted to $[-\pi/2, \pi/2]$. Its domain is $[-1, 1]$ and its range is $[-\pi/2, \pi/2]$.
• Arccosine comes from the cosine function restricted to $[0, \pi]$. It has the same domain, and its range is $[0, \pi]$.
• Arctangent comes from the tangent function restricted to $(-\pi/2, \pi/2)$. It is defined for every real number, and its values approach the endpoints of its range as $x$ grows without bound.
• Arccotangent uses the interval $(0, \pi)$ and is defined on all of $\mathbb{R}$.
• Arcsecant and arccosecant are the least uniform. Their principal ranges are usually given as $[0, \pi]\setminus\{\pi/2\}$ and $[-\pi/2, 0)\cup(0, \pi/2]$, but some texts choose other intervals, so the convention should be checked against the source being used.

These intervals are conventions rather than logical necessities. They are chosen so that the principal value is continuous across the domain and agrees with the familiar values near zero. For example, $\arcsin 0 = 0$ and $\arccos 0 = \pi/2$. Because the principal branches are smooth on the open intervals, the inverses have simple derivatives: $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}$ on $(-1,1)$, and $\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$.

Notation

The two common notations are $\arcsin x$ and $\sin^{-1} x$. The "arc" prefix reflects the geometry of the unit circle, where an angle of $\theta$ radians corresponds to an arc of that length. Lagrange popularised the arc names in the 1770s, and the superscript form, introduced by John Herschel in 1813, was modelled on operator notation in calculus. Because $\sin^{-1} x$ is easily confused with the reciprocal $(\sin x)^{-1} = \csc x$, many authors prefer $\arcsin x$, while calculators and programming languages commonly use the abbreviations $\text{asin}$ or $\text{atan}$.

Solving trigonometric equations

A principal inverse returns only one answer, so it cannot by itself solve an equation with several solutions. The equation $\sin\theta = 1/2$ has the principal solution $\theta = \pi/6$, but using the symmetry of the sine curve, the full set of solutions is $\theta = \pi/6 + 2k\pi$ and $\theta = 5\pi/6 + 2k\pi$ for integers $k$. Similarly, $\cos\theta = c$ has solutions $\theta = \pm\arccos c + 2k\pi$. Care is also needed when simplifying: $\arcsin(\sin x) = x$ holds only for $x$ in $[-\pi/2, \pi/2]$. When a problem asks for solutions in a given interval, such as $[0, 2\pi)$, the principal value is used as a starting point and the other solutions are found from symmetry and periodicity. For instance, $\tan\theta = 1$ on $[0, 2\pi)$ gives $\theta = \pi/4$ and $\theta = 5\pi/4$.

The two-argument arctangent

The quotient $y/x$ cannot distinguish between points such as $(1,1)$ and $(-1,-1)$, since both give the same ratio. The function $\operatorname{atan2}(y, x)$ avoids this problem by taking both coordinates. It returns the angle from the positive $x$-axis to the ray through the point $(x, y)$, with the quadrant determined by the signs of the two inputs. Its usual principal range is $(-\pi, \pi]$, and it handles $x = 0$ as a special case instead of dividing by zero. The function is widely used in polar coordinate conversion, in computing the argument of a complex number, and in robotics and graphics software, where directions must be recovered from vector components.

Inverse functions in integration

Inverse trigonometric functions appear as antiderivatives of common algebraic expressions. Among the standard results are $\int \frac{dx}{\sqrt{1-x^2}} = \arcsin x + C$ and $\int \frac{dx}{1+x^2} = \arctan x + C$. In trigonometric substitution, an expression such as $\sqrt{a^2 - x^2}$ is simplified by setting $x = a\sin\theta$, and the inverse function $\theta = \arcsin(x/a)$ is then used to convert the result back into the original variable. These techniques are used in calculus to evaluate integrals and to find the angles that determine geometric quantities.

Power series and Euler's formula

Taylor series of sine and cosine

The sine and cosine functions can be written as infinite polynomials, which is the form of the definitions that the rest of the theory depends on. The starting point is the power series of the exponential function, $e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n!}$. Sorting its terms by parity gives the two series. Even powers combine into cosine and odd powers into sine:

The derivation of Euler's formula involves concepts from calculus, power series, and complex numbers. These expansions are special cases of the Taylor series of a function at a point. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, which is the case for both functions here. Truncating the sine series gives a practical approximation. For small arguments, a few terms are enough, and the error can be bounded by a term of the next degree.

Extension to complex arguments

Because the series contain no reference to real angles, they can be used unchanged for complex inputs. The complex sine and cosine functions are defined using the same series expansions as the real cosine and sine functions, except that the inputs are allowed to be complex. The series have infinite radius of convergence, so they converge for all complex values of z. As a result, the functions are holomorphic everywhere on the complex plane. They are analytic functions in the strict sense, and they are entire functions (to be more precise, entire transcendental functions).

The series also fix the parity of the functions. Sine is an odd function and cosine an even function. The exponential form follows from the same expansions, with cosine and sine written as combinations of $e^{iz}$ and $e^{-iz}$:

Euler's formula and its consequences

Substituting $ix$ for $z$ in the exponential series and separating real and imaginary terms gives the result that is now called Euler's formula:

The argument rests on the fact that the exponential series converges absolutely, so its terms can be rearranged without changing the sum. Grouping the real and imaginary terms together then yields the decomposition into cosine and sine. The formula is named after Leonhard Euler, and it establishes the fundamental relationship between the trigonometric functions and the complex exponential function. The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case.

Several consequences follow directly. Setting $x=\pi$ gives $e^{i\pi}=-1$, known as Euler's identity, which links five constants of mathematics in one equation. The addition formulas for sine and cosine also follow from the rule $e^{a}e^{b}=e^{a+b}$. Multiplying $(\cos a+i\sin a)$ by $(\cos b+i\sin b)$ and comparing real and imaginary parts recovers the addition formulae of sine and cosine. Euler's formula therefore turns many trigonometric identities into simple rules for exponentials, which is why it is used throughout complex number theory and in signal processing.

Analytic properties on the complex plane

Because the sine and cosine are entire, the tools of complex analysis apply to them. Their values off the real axis behave differently from their real behaviour. For example, $\sin(iy)=i\sinh y$ grows without bound as $|y|$ increases, so the complex sine is unbounded, while on the real line it stays between −1 and 1. Its zeros are still isolated and occur only at integer multiples of $\pi$. The fact that these functions are entire is what allows the exponential form and the identities above to hold on the whole plane, not just on the real line.

Characterizations by differential and functional equations

Sine and cosine from a differential equation

In analysis, the two functions can be introduced without any appeal to triangles or circles. The exponential function, power series, and differential equations each provide a route, and these routes work for complex inputs as well without reference to any geometric notions.

The differential route starts from $y'' = -y$. Two conditions at the origin are enough to isolate each function: fixing $y(0)=0$ and $y'(0)=1$ selects sine, while $y(0)=1$ with $y'(0)=0$ selects cosine within the 2-dimensional function space V consisting of all solutions of this equation. The pair spans the whole two-dimensional family, so every solution is a combination $c_1\cos x + c_2\sin x$ whose coefficients are set by the starting values. Existence and uniqueness theory for linear equations guarantees that these initial-value problems have exactly one answer. When that answer has a Taylor expansion at the origin, its coefficients are forced by the starting data, which is how the power series of sine and cosine can be recovered from the equation alone.

Because both functions satisfy the same equation, each one is an eigenfunction of the second-derivative operator. This is the mathematical reason sine and cosine appear wherever a linear system returns to its own shape, from springs and pendulums in classical-mechanics to linear operators in general. Trigonometric functions arise as solutions of linear as well as partial differential equations, which is why they show up in the study of partial differential equations as well.

Functional equations

A second family of characterizations uses functional equations instead of derivatives. One of them is the subtraction formula $\cos(x-y)=\cos x\cos y+\sin x\sin y$. A continuous pair of functions obeying this formula is forced to be sine and cosine once an additional normalizing condition is imposed the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula.

A related equation is $f(x+y)+f(x-y)=2f(x)f(y)$, which is now called d'Alembert's functional equation. It was put forward around 1750 and arose in modelling the motion of a stretched string. Cosine is one of its solutions, so the equation is also known as the cosine functional equation. Its continuous solutions form a short, explicit list. According to a recent classification, the solutions are the zero function, the constant 1, and scaled versions of cosine and hyperbolic cosine, $H(t) = \cos(\alpha t)$ with $\alpha>0$, and $H(t) = \cosh(\alpha t)$ with $\alpha>0$, with the branch determined by the sign of the second derivative at zero. An early study of a companion equation for sine appeared in 1910, when E. B. Van Vleck published A functional equation for the sine, Ann. Math. Second Ser. 11(4), 161-165 (1910).

Uniqueness and the link to exponentials

Uniqueness theorems are what make these characterizations usable. The complex exponential $e^{it}$ can be defined as the unique solution of the initial value problem y' - iy = 0; y(0) = 1. Since $\cos t + i\sin t$ solves the same problem, the two must coincide, which gives Euler's formula $e^{it} = \cos t + i\sin t$.

The same argument proves the addition theorem. Both $\sin(t+c)$ and $\sin t\cos c+\cos t\sin c$ solve $y''+y=0$ with identical data at zero, so they agree for all $t$ By uniqueness, these functions are equal for all t, which proves the identity. Many other identities can be obtained in this manner by writing down a suitable initial-value problem.

Finally, the uniqueness of analytic continuation explains why the analytic definitions are preferred to purely geometric ones. From a theorem in complex analysis, there is a unique analytic extension of this real function to the complex numbers. The geometric definitions, the differential characterizations, and the functional characterizations therefore pick out the same function, and the analytic versions are the ones that extend cleanly beyond the real line.

Calculus of trigonometric functions

Derivatives

The derivatives of the six trigonometric functions rest on the limit $\lim_{h\to 0}\frac{\sin h}{h}=1$ and the addition formulas. The basic pair is

The other four follow from these with the quotient rule and the Pythagorean identity: $\frac{d}{dx}\tan x=\sec^2 x$, $\frac{d}{dx}\cot x=-\csc^2 x$, $\frac{d}{dx}\sec x=\sec x\tan x$, and $\frac{d}{dx}\csc x=-\csc x\cot x$.

These formulas hold only when angles are measured in radians. The limit $\sin h/h\to 1$ is true in radian measure alone. If x is given in degrees, differentiation produces an extra factor $\pi/180$ in every formula. Radian measure is therefore what keeps the derivatives free of constants. The chain rule extends them to composite arguments: $\frac{d}{dx}\sin u=\cos u\,\frac{du}{dx}$. Inverse trigonometric functions have algebraic derivatives, such as $\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}$ and $\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$, which follow from the inverse function rule.

Basic antiderivatives

Reversing the derivative table gives the standard integrals $\int\sin x\,dx=-\cos x+C$, $\int\cos x\,dx=\sin x+C$, $\int\sec^2x\,dx=\tan x+C$, and $\int\sec x\tan x\,dx=\sec x+C$. Two less obvious results are $\int\tan x\,dx=-\ln|\cos x|+C$ and $\int\sec x\,dx=\ln|\sec x+\tan x|+C$. The second is usually obtained by multiplying the integrand by $\frac{\sec x+\tan x}{\sec x+\tan x}$, which makes the numerator the derivative of the denominator. These antiderivatives are then applied through the fundamental theorem of calculus to compute definite integrals.

Powers and products

For $\int\sin^m x\cos^n x\,dx$, the method depends on parity. If m or n is odd, one factor is split off and the remaining even power is converted with $\sin^2x+\cos^2x=1$; the leftover is then integrated with the substitution u = cos x or u = sin x. If both exponents are even, the half-angle identities $\sin^2x=\tfrac12(1-\cos 2x)$ and $\cos^2x=\tfrac12(1+\cos 2x)$ reduce the power. Integrals of the form $\int\sin mx\cos nx\,dx$ are handled with product-to-sum formulas, for example $\sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)]$. The same device is central to Fourier series coefficient calculations.

Repeated integration by parts produces a reduction formula for powers:

Applied to definite integrals over $[0,\pi/2]$, this gives the Wallis-type values; for instance $\int_0^{\pi/2}\sin^4x\,dx=\tfrac{3\pi}{16}$.

Substitution and rewriting

Trigonometric substitution removes square roots of quadratic expressions. For $\sqrt{a^2-x^2}$ one sets $x=a\sin\theta$; for $\sqrt{a^2+x^2}$, $x=a\tan\theta$; for $\sqrt{x^2-a^2}$, $x=a\sec\theta$. As an example, $\int\sqrt{1-x^2}\,dx$ with $x=\sin\theta$ becomes $\int\cos^2\theta\,d\theta=\tfrac12\theta+\tfrac14\sin2\theta+C$, which returns to $\tfrac12\arcsin x+\tfrac12x\sqrt{1-x^2}+C$. Integration by parts handles products of polynomials and trigonometric functions; for example $\int x\sin x\,dx=-x\cos x+\sin x+C$. The Weierstrass substitution $t=\tan(x/2)$, with $\sin x=\frac{2t}{1+t^2}$, $\cos x=\frac{1-t^2}{1+t^2}$, and $dx=\frac{2\,dt}{1+t^2}$, converts any rational expression in sine and cosine into a rational integral. Many integrands are simpler after an identity is applied first, such as $\int\frac{dx}{1+\cos x}=\tan\frac{x}{2}+C$.

Arc length, area, and volume

For a curve $y=f(x)$ on $[a,b]$, the arc length is $L=\int_a^b\sqrt{1+f'(x)^2}\,dx$. For the cycloid $x=r(t-\sin t)$, $y=r(1-\cos t)$, the speed simplifies to $\frac{ds}{dt}=2r\left|\sin\frac t2\right|$, so one arch from $t=0$ to $t=2\pi$ has length $8r$. The area between $y=\cos x$ and $y=\sin x$ on $[0,\pi/4]$ is $\int_0^{\pi/4}(\cos x-\sin x)\,dx=\sqrt2-1$. For volumes of revolution, the disk method applied to $y=\sin x$ on $[0,\pi]$ about the x-axis gives $V=\pi\int_0^\pi\sin^2x\,dx=\frac{\pi^2}{2}$.

Fourier series and periodic functions

A function $f$ with period $2\pi$ can often be written as a constant term plus an infinite sum of harmonics, each a sine or cosine whose frequency is a whole-number multiple of the fundamental. In the standard form,

and the coefficients are found by integrating over one period:

The idea grew out of heat conduction. Joseph Fourier worked on the problem between 1807 and 1811, and his Théorie analytique de la chaleur appeared in 1822. There he proposed that an arbitrary function could be expanded in trigonometric series, and he used separation of variables to solve the heat equation, a method that is now standard in undergraduate courses. Whether every function admits such an expansion, and in what sense the sum returns the function, took much of the nineteenth century to settle.

The coefficient formulas work because of orthogonality. Over a full period, the integral of $\sin mx \sin nx$ or $\cos mx \cos nx$ is zero when $m \ne n$ and equals $\pi$ when $m = n \ge 1$, while $\sin mx \cos nx$ integrates to zero for all integers $m, n$. Multiplying the series by one basis function and integrating therefore cancels every other term and isolates a single coefficient. This is the same pattern as an inner-product in a vector space: the sines and cosines form an orthogonal family, and the Fourier coefficients are projections onto its members. The same logic underlies the discrete-fourier-transform for sampled data, where the continuous integrals become finite sums.

Two standard examples show how quickly the coefficients reveal a signal's structure. A square wave that equals $1$ on $(0,\pi)$ and $-1$ on $(-\pi,0)$ has only sine terms and only odd harmonics. Its coefficients are $b_n = 4/(n\pi)$ for odd $n$ and zero for even $n$, giving

A sawtooth wave with $f(x) = x$ on $(-\pi,\pi)$ is odd, so all cosine coefficients vanish. Its sine coefficients are $b_n = 2(-1)^{n+1}/n$, so

In both cases the amplitudes fall off like $1/n$, which reflects the jumps: the slow decay is the price of representing a discontinuity with smooth sinusoids. Smooth functions have coefficients that decay much faster, so their series converge more quickly.

Convergence is the subtle part. For a piecewise smooth periodic function, the series converges at every point of continuity and, at a jump, to the average of the one-sided limits. Near the jump, however, the partial sums do not settle down uniformly. They overshoot the higher side and undershoot the lower side, and adding more terms moves the oscillations closer to the discontinuity without reducing their height. The peak overshoot tends to about 9% of the full size of the jump. This behaviour was noticed by H. Wilbraham in 1848, and J. Willard Gibbs described it in letters to Nature in 1898 and 1899, which is why it bears his name as the gibbs-phenomenon. Its effects are important in signal processing and imaging, where truncated series produce ringing at sharp edges. Remedies include smoothing the partial sums by Fejér or Lanczos weights, which removes the overshoot at the cost of some sharpness. In the mean-square sense, by contrast, the series converges for any square-integrable periodic function, so the overshoot is a feature of uniform, not average, behaviour.

The sine and cosine functions in this expansion are the same ones that underlie the rest of this article; see harmonic-analysis for the broader theory of decomposing functions into frequency components, and heat-equation for the partial differential equation that motivated Fourier's original work.

Ancient and medieval origins

Trigonometry developed mainly as a tool for astronomy. The Greeks did not work with the sine as it is used today. Their only trigonometric quantity was the chord, which is closely related to the sine. The sole trigonometric function used by the ancient Greeks is the chord, which is closely related to the sine function. The first chord table is credited to Hipparchus in the second century BCE. The table itself is lost, but later reconstructions suggest that he treated every triangle as inscribed in a circle, so that each side became a chord. He considered every triangle as being inscribed in a circle, so that each side became a chord. Reconstructions also suggest that the circle was divided into 360 degrees and that intermediate values were found by linear interpolation. The table was based on a circle divided into 360 degrees with each degree divided into 60 minutes.

The most complete surviving Greek treatment is in Ptolemy's Almagest, written in the second century CE. Ptolemy's table gave chord lengths for angles ranging from 1⁄2 to 180 degrees by increments of 1⁄2 degree. His method relied on a geometric theorem about inscribed quadrilaterals, and some of the relations he derived are equivalent to the modern sum and difference formulas for sine. With a different interpretation of a and b, the sum formula results: These, of course, correspond to the sum and difference formulas for sines. Ptolemy's chord values were given to two sexagesimal places, and the work was known for its precision. Greek trigonometry was largely presented in geometric language rather than as a set of named ratios, and the modern symbols were not introduced until the seventeenth century.

In India, the tradition shifted from chords to half-chords, which are the sine in the modern sense. Aryabhata's Aryabhatiya, composed in 499 CE, is the first known text to name the sine and to give a table of it. The first mention of the sine function is to be found in his (one and only) seminal work the Aryabhatiya (499 CE). He called it jya, or half-bowstring, and introduced related terms such as the cosine and the versine. He constructed the first known sine table, introduced the terms jya (sine), kojya (cosine), and utkrama-jya (versine), and developed a remarkable difference equation for computing successive sine values. The table was computed at steps of 3°45′, and Aryabhata obtained the intermediate values by differencing. This entails taking the difference of the sine of two closely spaced angles and then taking the second sine difference. Later Indian work refined this approach. Brahmagupta devised a second-order interpolation formula that is equivalent to a second-order Taylor approximation, which allowed the sine of an arbitrary angle to be estimated more accurately.

Islamic mathematicians working in Baghdad and nearby centers rebuilt trigonometry around the sine and added new functions. al-Battani, active around 850 to 929 CE, abandoned the Greek chord system. He gave up the old Greek system of chords of angles and adopted the far more convenient trigonometrical proportion known as the sine; he also used its converse ratio, the cosine. He also used trigonometry on spherical projections for astronomy, and he is often credited with the first use of the names "sine" and "cosine". Abu al-Wafa' (940–998), working in Baghdad, extended the subject further. He is credited with introducing the secant and cosecant functions, as well studying the interrelations between the six trigonometric lines associated with an arc. He also computed sine and tangent tables at intervals of 15 minutes of arc, and he proved the familiar addition theorem for sines, which is a lot more efficient and elegant as compared to the original statement in Almagest involving chord lengths. His systematic use of spherical trigonometry was applied to astronomical problems, and al-Biruni later used the sine in astronomy and geography. Sources disagree on the tangent: Indian mathematicians had already described it, and Abu al-Wafa' is credited with showing how it could be used in calculations.

These results reached Europe through Latin translations of Arabic and Greek texts, mainly from the twelfth century onward, with centers such as Toledo playing an important role. Ptolemy's Almagest and the Arabic astronomical tables that built on it became standard references in medieval European universities. The transmission was gradual, and the Latin scholars who received these texts often worked from translations of translations. Their adoption of the trigonometric material laid the groundwork for the independent discipline that developed in Europe in the fifteenth century.

Early modern development and notation

For most of its history, trigonometry in Europe was a tool of astronomy. The shift toward an independent discipline is usually traced to Regiomontanus (Johann Müller, 1436–1476). He recognized that astronomers needed a systematic method for solving triangles, and he promised such a treatise in the introduction to his astronomical work. His De triangulis omnimodis was compiled decades earlier than De revolutionibus, in 1464, but it was not published until 1533. Its importance lay less in new results than in its organization. It reoriented trigonometry away from its primary connection with astronomy and toward a place within mathematics, specifically the Euclidean geometry of triangles and circles. The work was also structured like Euclid's Elements, and its first book set out definitions including the sine. Regiomontanus also reframed the subject as the study of solving triangles, and both On Triangles and modern textbooks introduce the sine by means of right triangles. The law of sines appears in a form similar to its modern expression. Regiomontanus omitted tangents from this book and supplied them in a separate astrological work, the Tabulae Directionum. The treatise was edited by Johannes Schöner and finally published in Nürnberg in 1533 by Johannes Petreius, together with Peuerbach's sine table, becoming a standard reference work for much of the next century.

In the late sixteenth century, François Viète worked on trigonometry with the same algebraic outlook that shaped his symbolic algebra. He gave formulas expressing $\sin nx$ in terms of $\sin x$ and $\cos x$, and he derived identities used to compute his sine tables. One such identity has since been given a visual proof. Viète's vocabulary was also experimental: he called the cosine the sinus residuae, and he used the terms amsinus and prosinus for related quantities. Names for the other functions were also settled in this period. The name tangent was first used by Thomas Fincke in 1583. The term cotangens was first used by Edmund Gunter in 1620. Abbreviations followed later, and cot was first used by Jonas Moore in 1674.

The reciprocal functions came into use later and for practical reasons. The secant and cosecant were not used by the early astronomers or surveyors. These came into their own when navigators around the 15th Century started to prepare tables. Copernicus had already used the secant, which he called the hypotenusa. In modern notation the reciprocal relations are $\sec A = \frac{1}{\cos A}$ and $\csc A = \frac{1}{\sin A}$, together with $\cot A = \frac{1}{\tan A}$. The full set of six functions, with standardized names and abbreviations, became common in the textbooks of the eighteenth and nineteenth centuries.

Leonhard Euler's Introductio in analysin infinitorum (1748) marks the point at which the functions moved from geometry into analysis. Euler's Introductio is considered the beginning of modern trigonometry. The book also made the function the central concept of analysis, and Euler introduces the notation f(x) for a function. Euler treated the sine and cosine through infinite series and did not rely on a triangle-based definition. He takes the limit of a very small arc and a very large number of steps, which gives the Taylor series for sine and cosine. The same analysis produced his relation between the complex exponential and the circular functions. Euler, in 1748, gave the formula exp(ix) = cos x + i sin x. This connection between the symbols for angles and the symbols for real numbers became central to later work with complex numbers. Euler's notation and definitions fixed the conventions that most modern texts still use.

Surveying, navigation, and astronomy

Surveying was one of the first practical uses of trigonometry. If the angles of a triangle and the length of one side are known, the remaining sides follow from trigonometric ratios. If the angles of a triangle and the length of one of its sides are known, the lengths of the remaining sides can be calculated using basic trigonometric functions. Surveyors therefore measure one carefully laid baseline on the ground and compute the rest of a network of triangles, which is far easier than walking every distance. Gemma Frisius described this procedure in 1533 as a way to divide a region into triangles from known baselines, and Willebrord Snell extended a chain of triangles across roughly 130 kilometers from Alkmaar to Bergen op Zoom, with his method published in 1617. Jean Picard later used the technique to measure a one-degree arc of meridian north of Paris in 1669–1670. In the following centuries, national networks were built on the same principle. The Principal Triangulation of Great Britain was begun by the Ordnance Survey in 1783, though not completed until 1853, and the Great Trigonometrical Survey of India, which ran from 1802 to 1871, became one of the largest mapping projects of its era.

Because the Earth is curved, the flat-plane version of these calculations has limits. For small networks covering areas of a few square miles, plane trigonometry suffices, treating the Earth as flat and using Euclidean geometry. Large geodetic networks instead require spherical trigonometry, which treats triangles drawn on the surface of a sphere. The same framework underlies navigation. A position on the sea is fixed by angles to celestial bodies, and the shortest route between two points on Earth lies along a great circle. Its length and bearings are computed with the spherical law of cosines and related formulas, so that ships and aircraft can plan courses that are much shorter than rhumb lines drawn on a flat chart.

The marine sextant gave navigators a practical way to take those angles. In 1731, John Hadley, an Englishman, and Thomas Godfrey, in Philadelphia, independently developed instruments for measuring angles by means of double reflection. Hadley's design was an octant, an eighth of a circle, and it was a simple matter to enlarge Hadley's octant, an eighth of a circle, to the sextant, a sixth of a circle, that could measure up to 120 degrees. The instrument determines the altitude of a celestial body above the horizon or the angular distance between two celestial bodies. The observed angles were converted to latitude and, when combined with timekeeping, longitude. A sextant can also be used to measure the lunar distance between the moon and another celestial object to determine Greenwich Mean Time and hence longitude. Isaac Newton had sketched a similar reflecting instrument around 1699, but it remained unpublished until his papers were examined later. Sextants were also used on land for surveying, and in 1922 Carlos Viegas Gago Coutinho adapted the instrument for aeronautical navigation.

In astronomy, trigonometry measures distances that cannot be reached directly. The principle of stellar parallax is that a nearby star appears to shift against distant background stars as Earth moves around the Sun, and the size of that shift gives the distance. Friedrich Wilhelm Bessel used the Earth's orbit as a baseline. In 1838, Bessel announced that 61 Cygni had a parallax of 0.314 seconds of arc, placing it at a distance of three parsecs, or ten light-years. Modern measurements place the star at about 11.4 light-years. Thomas Henderson published a parallax for Alpha Centauri in 1839, derived from observations made earlier at the Cape of Good Hope. Parallax distances are expressed simply: a star with parallax $p$ arcseconds lies at $1/p$ parsecs. The same spherical geometry is used to convert measured positions on the celestial sphere into coordinates, and to predict the orbits of planets and moons, including their positions relative to the Sun and Moon.

Older instruments such as the astrolabe embodied the same geometry in a portable form. Its scales could be read to find the altitude of a star or the Sun, and the observed value could be used to estimate time or latitude with the aid of trigonometric reasoning. Although modern satellite positioning has largely replaced these methods on land and sea, the trigonometric relationships behind triangulation, spherical navigation, and parallax remain the foundation of geodesy and satellite positioning.

Physics: oscillations, waves, and rotation

Sine and cosine appear throughout physics because many systems return to equilibrium with a restoring force proportional to displacement, which yields a linear second-order differential equation. For a mass on an ideal spring, $m\,x'' = -kx$, and the solutions are $x(t)=A\cos(\omega t+\varphi)$ with $\omega=\sqrt{k/m}$. The amplitude $A$ and phase $\varphi$ are fixed by the initial position and velocity. A pendulum follows the same pattern for small swings. The equation $\theta'' = -(g/L)\sin\theta$ becomes linear when $\sin\theta\approx\theta$, giving $\omega=\sqrt{g/L}$. For large amplitudes the period departs from this, and its exact form requires elliptic integrals, which are treated in the generalizations section.

Waves are described the same way. A sinusoidal disturbance $u(x,t)=A\sin(kx-\omega t+\varphi)$ satisfies the one-dimensional wave equation $\partial^2 u/\partial t^2 = c^2\,\partial^2 u/\partial x^2$ when $c=\omega/k$, where $c$ is the propagation speed, $k=2\pi/\lambda$ is the wavenumber, and $\omega$ is the angular frequency. In 1747 d'Alembert showed that the general solution splits into a part moving left and a part moving right, and sinusoids are a convenient building block for both. Sound in a pure tone is a pressure variation of this form, and in Maxwell's electromagnetic theory of 1865 light is a transverse wave whose electric field oscillates sinusoidally in space and time. Because a general periodic signal can be decomposed into such components, the sinusoid becomes the basic unit for the analysis developed in the Fourier and signal-processing sections.

Optics gives the most familiar trigonometric law in wave physics. Snell's law, $n_1\sin\theta_1 = n_2\sin\theta_2$, relates the angles of incidence and refraction to the refractive indices of two media. Willebrord Snellius formulated it in 1621, and it can be derived from Fermat's principle of least time. Its sine form follows from the geometry of wavefronts: the ratio of the sines equals the ratio of the phase speeds in the two media. Diffraction is also governed by sines. A plane wave passing through a slit of width $a$ has its first minimum at $a\sin\theta=\lambda$, and a diffraction grating with spacing $d$ produces bright maxima where $d\sin\theta = m\lambda$. In 1913 William Henry Bragg and William Lawrence Bragg showed that X-rays reflected from successive crystal planes interfere constructively when $n\lambda = 2d\sin\theta$. That relation, derived from the extra path length $2d\sin\theta$, became the basis of X-ray crystallography.

Rotation supplies the geometric origin of these waves. A point moving at constant angular velocity $\omega$ around a circle of radius $r$ has coordinates $(r\cos\omega t,\ r\sin\omega t)$. Its projection onto either axis is a sinusoid, so uniform circular motion and simple harmonic motion are two views of the same process. This is why the phase of an oscillation can be treated as an angle. Radian measure makes the relations simple: arc length is $s=r\theta$, and angular velocity $\omega=d\theta/dt$ is measured in radians per second. A rigid body rotating through an angle $\theta$ is described by a rotation matrix built from $\cos\theta$ and $\sin\theta$, which links these physical rotations to the matrix methods used in later sections.

The same mathematics governs forced systems. When a damped oscillator is driven sinusoidally, its steady-state response oscillates at the driving frequency, and its amplitude peaks near the natural frequency in a phenomenon called resonance. Two waves of identical frequency combine into a single sinusoid whose phase is the sum of their phases, which is the physical basis of the harmonic relationship between sine and cosine described elsewhere in this article.

Electrical engineering and telecommunications

Alternating current is one of the main places where sine and cosine do practical work. The voltage from a rotating generator varies sinusoidally in time, so it can be written as $v(t)=V_m\cos(\omega t+\alpha)$, where $V_m$ is the peak amplitude, $\omega$ the angular frequency and $\alpha$ the phase. Engineers seldom manipulate such expressions directly. The phasor method replaces each sinusoid with a single complex number, $V_m e^{j\alpha}$, that keeps only amplitude and phase and suppresses the common factor $e^{j\omega t}$ (Phasor voltage is defined as a complex number representation of voltage in an alternating current circuit). Geometrically, the phasor is a vector of fixed length that turns at constant angular velocity, and its vertical projection gives the instantaneous value of the wave (A phasor is a vector rotating at a constant angular velocity.; At any angle θ, OA sinθ equals the instantaneous value of the sine wave.). Once the problem is recast as complex arithmetic, time derivatives become multiplication by $j\omega$, and resistors, capacitors and inductors acquire impedances that combine in the same way as resistances. For a resistor and capacitor in series, the phasor current is $I = V_{in}/(R + 1/j\omega C)$ (We can solve the circuit for the phasor current I, using Ohm's Law). The same machinery expresses the phase difference between voltage and current, which is the basic quantity in power calculations.

Telecommunications rests on sinusoidal carriers. A high-frequency sine wave is altered according to a lower-frequency message, called the modulating or baseband signal. Three properties can be varied: amplitude, frequency and phase (Modulation refers to the technique of applying variations to some operating characteristic of a sinusoidal signal (amplitude, frequency, or phase).). In amplitude modulation (AM), the carrier's envelope follows the message while its frequency stays fixed (When the modulating information signal (a sine wave) is applied, the carrier amplitude rises and falls in accordance. The carrier frequency remains constant during amplitude modulation.). In its simplest form the transmitted wave is the carrier multiplied by one plus the baseband signal (x_AM = x_C(1+x_BB)). Because multiplication in time corresponds to convolution in frequency, the message spectrum reappears as sidebands at sum and difference frequencies around the carrier (The modulating process results in three frequencies: a difference frequency between carrier and baseband signal, a sum frequency between carrier and baseband, and the carrier signal.). The ratio of message amplitude to carrier amplitude, the modulation index, describes the depth of modulation (Modulation Index m = MAX(V_m(t))/V_co). A drawback is that much of the power sits in the carrier, which carries no information (Simple amplitude modulation makes inefficient use of the transmitted power, as information is transmitted only in the sidebands but the majority of the power is contained in the carrier.).

In frequency modulation (FM), the amplitude is kept constant and the instantaneous frequency is shifted in proportion to the message (the instantaneous frequency of the carrier is made to increase or decrease from the carrier frequency by an amount proportional to the modulating or baseband signal.). AM was the dominant form of civilian broadcasting until the late 1970s, after which FM became dominant (Amplitude modulation (AM) is the earliest form of modulation that was adapted for civilian use, and it remained the dominant form of modulation in civilian radio broadcasting until the late 1970s. After that time, frequency modulation (FM) became dominant and AM radio began its slow decline.). Modern digital systems extend the same idea: quadrature amplitude modulation changes amplitude and phase together (In quadrature amplitude modulation (QAM), the amplitude and phase are changed.), and high-speed digital data is sent over high-frequency carriers (Frequency modulation and amplitude modulation are still with us to this day, but now these modulation methods are used to send digital data at high speeds over a high frequency carrier wave.).

Filter design depends on the response of linear circuits to sinusoids. A linear system driven by a sine wave returns a sine wave at the same frequency, changed only in amplitude and phase, and both changes are fixed by the transfer function evaluated at that frequency (The response of a LTI system to a sine wave is a sine wave at the same frequency. Amplitude and phase are determined by the transfer function's gain and phase at this frequency.). The simplest example is the RC low-pass filter. Substituting $j\omega$ for the complex frequency gives $H(j\omega)=1/(1+j\omega RC)$, with magnitude $1/\sqrt{1+(\omega RC)^2}$ (H(jω)=1/(1+jωRC); the magnitude of the transfer function is 1/√(1+(ωRC)²)). The cutoff frequency, where the output falls to about 70.7 percent of the input, is $f_c = 1/(2\pi RC)$ (The cutoff frequency, defined as the frequency at which the output drops to 1/√2 of the input (approximately -3 dB)). The phase lag of such a circuit grows with frequency and approaches 90° at high frequency (As you approach the corner frequency the phase begins to lag and at high frequency it reach 90°.). Because a general signal can be regarded as a sum of sinusoids, its spectrum serves as a recipe for the signal (Given that any signal can be thought to be made up of sinusoidal signals, the spectrum then represents the "recipe card" of how to make the signal from sinusoids.), and the filter's frequency response determines how each component is scaled and shifted. Frequency-response plots of this kind are usually drawn as Bode plots (For our example RC circuit with R=10kΩ and C=47nF the Bode plot of the transfer function is shown on Figure 2.).

Communication standards are built on these principles. They fix carrier frequencies, channel bandwidths and modulation schemes so that many transmitters can share a band without interference. Carriers are chosen far above the message band; a typical rule of thumb requires the carrier to be at least ten times higher than the baseband signal (It is typically a radio frequency (RF) wave and must be much higher in frequency (at least by a factor of 10) than the modulating baseband signal.). Radio stations, for example, are assigned different carrier frequencies so that they can coexist on the same medium (assigned a different carrier frequency (like radio stations) and still share the same channel.). Analysing these systems, from the design of a receiver's tuned circuits to the spectrum of a transmitted signal, relies on the sinusoidal steady-state methods described above. The same mathematics governs the 50 or 60 Hz alternating current of the electrical grid, which is why sinusoidal analysis is taught as a foundation of electrical engineering.

Digital signal processing and imaging

Once a physical signal is recorded as a finite list of numbers, sines and cosines become practical computing tools. The discrete Fourier transform (DFT) rewrites such a sample sequence as a sum of sinusoids at evenly spaced frequencies. Evaluated term by term, it requires on the order of n² multiplications for n samples, which was impractical for large data sets until the fast Fourier transform (FFT) changed the cost. The 1965 paper by James Cooley and John Tukey lowered the operation count to roughly n log n, and this single change made spectral analysis routine on digital computers. Their timing mattered as much as the algorithm: the method appeared just as digital machines were becoming widely available Although discovered previously, it was Cooley and Tukey's timing, which coincided with the beginning of widespread use and availability of digital computers, that led to its success. Earlier versions of the same idea had surfaced before. The recursive divide-and-conquer structure of the FFT was anticipated by Carl Friedrich Gauss around 1805, who used it to interpolate the orbits of the asteroids Pallas and Juno from observational data. Gauss's account remained unpublished during his lifetime and appeared only in his collected works. Demand for fast spectral analysis came from practical fields, including seismology, crystallography, and sonar.

Compression is the second major application. The discrete cosine transform (DCT) expresses a finite sequence of data points as a sum of cosines at different frequencies. Its key property is energy compaction: for typical image blocks, most of the signal energy concentrates in a few low-frequency coefficients, so the many small high-frequency coefficients can be coarsely quantized or discarded. Nasir Ahmed conceived the DCT while working at Kansas State University and proposed it to the National Science Foundation in 1972, initially aiming at image compression. Ahmed developed a working DCT algorithm with his PhD student T. Natarajan and friend K. R. Rao in 1973, and they presented their results in a January 1974 paper. The DCT became the core of the JPEG still-image standard, first released in 1992. The same family of transforms appears in audio coding: a variant, the modified discrete cosine transform, underlies MP3, AAC, and Vorbis. Aggressive compression exposes the trade-off, because blocky artifacts can appear when the coefficients are quantized too heavily.

Medical imaging relies on the same mathematics in a different way. A CT scanner records X-ray attenuation along many straight lines through a slice, taken from many angles. The Fourier slice theorem links the one-dimensional transform of each projection to a line through the origin of the two-dimensional transform of the object. This relationship underlies filtered back-projection, the earliest and most often used CT algorithm. In that method, each projection is filtered, usually with a ramp-like filter applied through fast Fourier methods, and then smeared back across the image grid along the direction it was measured. Commercial X-ray CT systems commonly use FFT-based implementations of this procedure. Magnetic resonance imaging follows a related logic: the scanner samples data in a Fourier-domain space often called k-space, and an inverse FFT converts those samples into an image. Both fields depend on the trigonometric basis functions that define the transform.

Radar, sonar, and geophysical exploration use the same tools for a different task: estimating frequency content and timing from received echoes or seismic traces. Spectral peaks reveal the frequencies present in a return signal, phase differences between sensors indicate direction, and small frequency shifts can encode the motion of a target through the Doppler effect. Because the FFT makes these computations fast enough for real-time operation, sinusoidal analysis is embedded in the signal chains of most modern sensing systems.

Numerical evaluation and computer graphics

Computing the functions on digital hardware

Early digital machines had slow multipliers, so designers looked for ways to produce sines and cosines from simple operations. The best-known answer is CORDIC, short for COordinate Rotation DIgital Computer. It was developed at the aeroelectronics department of Convair to replace the analog resolver in the B-58 bomber's navigation computer. Jack Volder published the method in 1959 in the Proceedings of the Western Joint Computer Conference. Later, John Stephen Walther at Hewlett-Packard further generalized the algorithm, allowing it to calculate hyperbolic and exponential functions, logarithms, multiplications, divisions, and square roots.

CORDIC treats an angle as a rotation applied to a two-dimensional vector. The rotation is broken into many small steps, and it accomplishes this by decomposing the desired function into a series of minute rotations and shifts. Each step needs only shifts, additions, and a small table of constants, so the method avoids multiplication and division. After the steps finish, the vector's coordinates give the cosine and sine, scaled by a known gain factor. CORDIC converges on results with increasing precision the more iterations it is run. The method was popular in pocket calculators, and CORDIC is generally faster than other approaches when a hardware multiplier is unavailable (e.g., in a microcontroller based system), or when the number of gates required to implement the functions it supports should be minimized (e.g., in an FPGA).

General-purpose processors now include floating-point instructions for these functions, so software CORDIC is rare on desktops and servers. Most mathematical libraries instead evaluate polynomial or rational approximations on a reduced interval. Such approximations work well only on a limited range, so the argument must first be mapped into it. It is valid for angle values between 0 and π/2 only, but whatever the value of an angle, the calculation of its tangent can always be reduced to that of an angle between 0 and π/2, using trigonometric identities.

Argument reduction and floating-point accuracy

Because the functions are periodic, a library first subtracts a multiple of π/2 from the argument x, records the quadrant, and evaluates the kernel on the remainder. For moderate x, Cody–Waite reduction splits the constant into pieces so that the subtraction stays exact. For very large x, the difficulty is that the subtraction removes many leading bits. The classical method computes x − kπ, and about e bits will be cancelled in the subtraction x − (kπ), thus we need to compute kπ with a precision of at least e + n bits to get an accuracy of at least n bits for x'.

Mary Payne and Robert Hanek's radian reduction avoids storing an enormous constant. Their paper notes that for a CRAY-1, VAX, or the proposed double-extended IEEE format, the maximum argument which might be presented for reduction is of the order of 2^16000 radians. Accurate reduction of such an argument would require storage of 1/π (or its reciprocal) to over 16,000 bits. Their procedure computes only the bits of the constant that survive the reduction. Modern implementations follow the same idea. Payne-Hanek works by representing 1/P at very high precision (typically as a multiword fixed-point constant), multiplying x by that constant to obtain a high-precision quotient, extracting the nearest integer k, and then reconstructing the remainder. Most arguments encountered in practice are modest, so the expensive path is reserved for rare large inputs. Most common input arguments to the trigonometric functions are small (say, less than 8) or sometimes medium (say, between 8 and approximately 260).

Errors can be severe when reduction is done carelessly. In one open-source 256-bit implementation, sin(1e100) was 0.9318... instead of -0.3723... The argument was reduced by Mod(2Pi), and the approximation error of 2Pi was amplified by the argument. Accuracy also needs extra care near zeros of the function, where the reduced value is tiny and the relative error is large. Radian reduction allows efficient calculation of additional bits to maintain full accuracy in the neighborhoods of the zeros of the desired function. Near the poles of tangent, the same issue appears in a different form, since division by a very small cosine amplifies any error.

Rotations in computer graphics and robotics

Rotations of the plane are written with a matrix whose entries are cosines and sines of the angle, and rotations in space are built by multiplying such matrices. Rotation matrices are orthogonal, so their inverses are their transposes, and chains of joints in robot arms are composed this way. A 3D rotation stored as a matrix uses nine numbers, which is computationally intensive in practice for some applications.

A common alternative is to describe orientation by three Euler angles, applied in sequence about fixed axes, as in R = R_z(ψ)R_y(θ)R_x(φ). This parametrization is compact but singular. This approach, while geometrically intuitive, suffers from Gimbal Lock, a coordinate singularity that occurs when two of the three rotation axes become aligned, effectively reducing the system's degrees of freedom from three to two. For the Z–Y–X convention this happens when the pitch reaches ±90°.

Quaternions avoid this problem. A unit quaternion encodes a rotation about an axis u through angle θ as $q = \cos(\theta/2) + \sin(\theta/2)\,\mathbf{u}$, so trigonometric values appear only when the quaternion is constructed. Because they do not involve trigonometric functions, they are not subject to the instability encountered when the function approaches a discontinuity, which is why composing rotations is cheap. Quaternions also support smooth interpolation between orientations: Shoemake pioneered the use of quaternions in keyframe animation, demonstrating that SLERP effectively prevents gimbal lock during complex motions. Engines such as Unity often convert Euler-angle inputs into quaternions internally, which can be useful for more complex motions that might otherwise lead to gimbal lock.

Geometry in higher dimensions and complex analysis

Spherical trigonometry

When a triangle is drawn on a sphere rather than a plane, its sides are arcs of great circles, and each side can be measured by the central angle it subtends at the sphere's centre. For a triangle with sides a, b, c and the angle C opposite c, the spherical law of cosines takes the form $\cos c = \cos a\cos b + \sin a\sin b\cos C$. Taking C as a right angle gives the spherical Pythagorean relation $\cos c = \cos a\cos b$. The planar law of cosines is recovered as the sides shrink relative to the radius, so the spherical formula is the exact version of which the plane formula is an approximation for small figures. A second law, relating three angles to one side, follows from the polar dual triangle.

Numerical practice shows why the formula is often rewritten. Solving for a small side by inverting the cosine magnifies rounding error, so navigation software uses the haversine form, which is stable for nearby points. This is the standard method for great-circle distances between latitude and longitude pairs, and it underlies the calculation of geodesics on the Earth's surface for air and sea routes.

Rotations in higher dimensions

A rotation of Euclidean space $\mathbb{R}^n$ is a linear map represented by an orthogonal matrix of determinant 1. These matrices form the special orthogonal group $SO(n)$, in which composing rotations corresponds to multiplying matrices. As a Lie group, $SO(n)$ has dimension $n(n-1)/2$: one angle in the plane, three parameters in space (for which Euler angles give one coordinate system), and six in four dimensions.

The geometric description changes with dimension. In the plane, a rotation fixes only the origin and turns every other point through one angle. In three dimensions, a non-trivial rotation fixes a line, its axis, and turns the perpendicular plane. In four dimensions this axis picture breaks down, because every plane has a whole family of perpendicular planes, so the rotation is better described by its planes of rotation and an angle in each. The count of independent planes in dimension n is $\lfloor n/2 \rfloor$. Euler's rotation theorem generalises in the same way: in dimension $2n$ or $2n+1$, a rigid motion that fixes a point is a composition of at most n rotations in mutually orthogonal planes.

In applied linear algebra the simplest building block is the Givens rotation, which acts only in a single coordinate plane and leaves other coordinates unchanged. A suitable sequence of such rotations can zero chosen coordinates of a vector, which is how they are used in matrix factorisations and in the analysis of high-dimensional data. The same rotation matrices appear in the matrix formalism of linear maps and in the orthogonal matrices that preserve lengths and angles.

Complex trigonometric functions

The power series of sine and cosine converge for every complex argument, so they define holomorphic functions on the whole complex plane. They are entire functions, and because they are non-polynomial, the point at infinity is an essential singularity. Sine remains odd and cosine remains even, and the exponential form $\sin z = (e^{iz}-e^{-iz})/(2i)$ and $\cos z = (e^{iz}+e^{-iz})/2$ carries over unchanged from the real case.

Periodicity behaves differently off the real axis. Writing $z = x + iy$, the functions are $2\pi$-periodic in the real direction, but the terms involving $e^{y}$ and $e^{-y}$ never repeat as y changes, so no period exists in the imaginary direction. Consequently $|\sin z|$ and $|\cos z|$ grow like $e^{|y|}/2$ as $|y|$ becomes large. Since a bounded entire function must be constant by Liouville's theorem, and cosine is not constant, cosine is unbounded on the complex plane. Its only zeros are the real points $(2n+1)\pi/2$, so no complex zeros occur off the real axis.

Because the functions are periodic, their inverses are multivalued. The complex arcsine and arccosine are written through the complex logarithm and square root, and the square root produces branch points at $z=\pm 1$. Each branch of the inverse is then a holomorphic function on a slit plane, which is the framework behind the principal branches discussed in the inverse-functions section.

Conformal geometry in the complex plane

Using $\sin(x+iy) = \sin x\cosh y + i\cos x\sinh y$, one sees that a vertical line $x=\text{const}$ maps to an ellipse, and a horizontal line $y=\text{const}$ maps to a hyperbola, both with foci at $\pm 1$. The map $z\mapsto \sin z$ is locally angle-preserving, that is conformal, wherever its derivative $\cos z$ is non-zero. It fails to be conformal only at the critical points $(2n+1)\pi/2$ on the real axis, where it is locally two-to-one. This makes sine and cosine useful tools for mapping regions such as strips and half-planes in complex analysis and in the conformal transformations used in potential theory and fluid flow.

Generalizations and related functions

The circular functions belong to a larger family. Other families keep much of their algebra and analysis while replacing the circle, the angle, or the polynomial structure with something else.

Hyperbolic functions

The hyperbolic functions are built from the exponential function. Splitting $e^x$ into its even and odd parts gives the hyperbolic cosine and the hyperbolic sine, $\cosh x = \tfrac{1}{2}(e^x+e^{-x})$ and $\sinh x = \tfrac{1}{2}(e^x-e^{-x})$ (The even and odd parts of the function e^x are given the names of hyperbolic cosine and hyperbolic sine.). The geometric analogy is with the hyperbola $x^2-y^2=1$ rather than the circle. The point $(\cosh t, \sinh t)$ traces the right branch of this hyperbola, just as $(\cos t, \sin t)$ traces the unit circle ((cosh(t),sinh(t)) lies on the hyperbola x^2 - y^2 = 1, and in fact this is a parametrization of the hyperbola).

The parameter has a different meaning. Because a hyperbola does not wrap around the origin, a rotation angle cannot be used to walk along it, so the natural parameter is an area. Since the unit hyperbola doesn't wrap around the origin, it is easier for the hyperbolic trig functions to be parameterized by area. The algebra mirrors the circular case closely. The addition formulas for $\cosh$ and $\sinh$ resemble the trigonometric ones, but the sign in the sine term changes, and the derivatives are $\cosh' = \sinh$ and $\sinh' = \cosh$ (cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y) and sinh(x+y)=sinh(x)cosh(y)-cosh(x)sinh(y) ("addition formulas"), (cosh)'=sinh and (sinh)'=cosh).

Elliptic functions and the large-amplitude pendulum

Elliptic functions arise from a physical problem: the swing of a pendulum. For small swings the period is $T_0 = 2\pi\sqrt{L/g}$, independent of amplitude (T 0=2π√l/g, with l being the length of the pendulum and g the acceleration due to gravity.). For large swings the equation of motion is nonlinear, and the small-angle approximation no longer holds. The exact period can be written as a ratio involving a complete elliptic integral of the first kind, $T/T_0 = 2K(k)/\pi$, where $k$ depends on the amplitude (T/T 0=2K(k)/π, (2) where K(k) is the complete elliptic integral of the first kind). This integral has no elementary closed form, so the period is normally computed numerically (Because there is no known closed-form of K(x), the solution has to be computed numerically). The inverses of these integrals are the elliptic functions, which are doubly periodic and have many uses beyond pendulums (The inverses of elliptic integrals are known as elliptic functions, and they have numerous applications beyond that of pendulums.). Elliptic functions are the natural successors to sine in these nonlinear problems, and they are also classical examples of functions with two independent periods in the complex plane.

Chebyshev polynomials

The Chebyshev polynomials of the first kind connect directly to cosine. The degree-$n$ polynomial is defined by $T_n(x) = \cos(n \arccos x)$ for $-1 \le x \le 1$, which is equivalent to the identity $T_n(\cos\theta) = \cos n\theta$ (The Chebyshev polynomials of the first kind, denoted by T_n(x), are expressed as: T_n(x) = cos(n arccos (x)), where x ∈ [-1, 1]). Expanding $\cos n\theta$ in powers of $\cos\theta$ produces the polynomial, and a recurrence relation generates all the polynomials in turn, namely $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ (The Chebyshev polynomials of the first kind (T_n(x)){n≥0} and of the second kind (U_n(x)){n≥0} are defined by the recursions). Their zeros lie at $\cos\bigl((2k-1)\pi/2n\bigr)$, and this equioscillation property makes them useful in approximation theory, where they reduce the maximum error of polynomial approximations (These polynomials are widely used in approximation theory, particularly for minimizing the error in polynomial approximation.). Through the substitution $x=\cos\rho$, a Chebyshev series becomes an ordinary cosine series.

Other special functions

The same pattern appears in other settings. Bessel functions generalize sine and cosine to problems with cylindrical symmetry, such as vibrating drums and waves in pipes, and Legendre polynomials play a similar role for spherical geometry. These functions are often defined through differential equations or series, and they retain the orthogonality and oscillatory behavior that make the trigonometric functions useful in Fourier analysis.

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