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Trigonometric functions

Trigonometric functions are real-valued functions in mathematics that relate angles, typically in right-angled triangles, to the ratios of the lengths of the triangle's sides, providing a foundational tool for analyzing geometric relationships and periodic phenomena. The six principal trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—are defined as follows for an acute angle θ in a right triangle with opposite side o, adjacent side a, and hypotenuse h: sin θ = o/h, cos θ = a/h, tan θ = o/a, csc θ = h/o, sec θ = h/a, and cot θ = a/o. These functions extend beyond triangles to the unit circle, where sin θ and cos θ represent the y- and x-coordinates, respectively, of the point on the circle reached by rotating θ radians (or degrees) counterclockwise from the positive x-axis, enabling their application to any real . θ is then defined as the sin θ / cos θ, with the functions following accordingly. Key properties include their periodicity—sin and cos repeat every 2π radians, while tan repeats every π radians—and bounded ranges, such as -1 ≤ sin θ ≤ 1 and -1 ≤ cos θ ≤ 1 for all real θ. The development of trigonometric functions traces back to ancient astronomy and , with around 140 BC creating the first tables of lengths in circles, laying the groundwork for modern . Subsequent advancements by figures like , who established the identity sin²θ + cos²θ = 1, and Arab mathematicians such as Abu'l-Wafa, who derived formulas, evolved these from practical astronomical tools into systematic functions by the medieval period. By the 17th and 18th centuries, contributions from Euler and others standardized notation and integrated them with complex numbers and . In contemporary and , trigonometric functions are indispensable for modeling oscillatory behaviors, such as in physics and signals in , as well as in for derivatives and integrals, linear algebra for rotations, and statistics for periodic . They also underpin real-world applications, including for determining positions at sea, for and building design, and astronomy for calculating celestial distances.

Introductory Concepts

Notation

The primary trigonometric functions are denoted using the following standard symbols: sine by \sin, by \cos, by \tan, by \cot, by \sec, and by \csc. These abbreviations, shortened from their full names (such as "sine" from Latin and "" from the geometric line), have become the conventional notation in mathematical texts since the 17th century. The relationships are implicit in the notation, with \csc as the of \sin, \sec of \cos, and \cot of \tan. Historically, trigonometric notation evolved from earlier forms using full words or alternative abbreviations, such as "S" for or "R" for the in chord-based computations, to the more compact modern symbols introduced in the 1600s. For instance, the abbreviations "" and "" were introduced by around 1620–1624, while "" dates to Thomas Fincke's 1583 text; further popularized functional notation like \sin x in the late , treating these as functions of a rather than fixed ratios. This shift from verbose or geometric-specific symbols to concise, function-oriented ones facilitated the integration of trigonometry into and . The argument of a trigonometric function, which represents the input angle or value, is typically denoted by \theta (theta) in geometric and introductory contexts to highlight its angular interpretation, often in conjunction with units like degrees or radians. In contrast, x is the standard variable for arguments in broader mathematical applications, including real or complex numbers, allowing trigonometric functions to be analyzed as part of general function theory. Multi-angle notations, such as \sin(2x) or \cos(n\theta) where n is an , extend the basic function symbols to express compositions or multiples of the argument, enabling compact representation of periodic extensions and laying the groundwork for exploring relationships among trigonometric values. These forms underscore the functions' versatility in modeling oscillations and waves without altering the core symbolic conventions.

Angle Measurement

Angles are measured using two primary units: and radians. The , denoted by the °, is defined such that one complete rotation around a corresponds to 360 degrees. This unit traces its origins to ancient , where the (base-60) system led to the division of the full —or the path of —into 360 equal parts for tracking celestial movements. In contrast, the provides a more geometrically natural measure of angles. One is the subtended at the center of a by an whose length equals the circle's . Consequently, a full around the measures exactly $2\pi radians, linking angular measure directly to the circle's . To convert between these units, the formula radians = degrees \times \frac{\pi}{180} is used, reflecting the proportional relationship between the two systems. This conversion is essential for applications spanning and . Radians hold particular advantages in calculus, where trigonometric functions are differentiated and integrated. For instance, the derivative of \sin(x) is \cos(x) precisely when x is measured in radians, avoiding extraneous scaling factors that arise with degrees. This property simplifies many analytical computations and aligns angular measures with linear dimensions in a unit circle.

Right Triangle Definitions

The trigonometric functions can be defined geometrically using the ratios of the sides of a right-angled triangle, where one angle is exactly 90° and the other two angles are acute. In such a triangle, the side opposite the right angle is called the hypotenuse, which is the longest side, while the other two sides are the legs: one adjacent to the acute angle of interest and the other opposite to it. Consider an acute angle \theta in a right triangle, with the opposite side of length a, the adjacent side of length b, and the hypotenuse of length c. The sine function is defined as the ratio of the opposite side to the hypotenuse: \sin \theta = \frac{a}{c}. The cosine function is the ratio of the adjacent side to the hypotenuse: \cos \theta = \frac{b}{c}. The tangent function is the ratio of the opposite side to the adjacent side: \tan \theta = \frac{a}{b}. The remaining three trigonometric functions are the reciprocals of these: the cosecant is \csc \theta = \frac{1}{\sin \theta} = \frac{c}{a}, the secant is \sec \theta = \frac{1}{\cos \theta} = \frac{c}{b}, and the cotangent is \cot \theta = \frac{1}{\tan \theta} = \frac{b}{a}. These definitions apply specifically to acute angles \theta in the , so $0^\circ < \theta < 90^\circ (or $0 < \theta < \frac{\pi}{2} in radians), where all ratios are positive. To visualize, imagine a right triangle with the right angle at vertex C, acute angle \theta at vertex A, opposite side a (from B to C), adjacent side b (from A to C), and hypotenuse c (from A to B); the labels align with the standard ratios above. A common mnemonic for recalling the definitions of sine, cosine, and tangent is SOH-CAH-TOA, where SOH stands for "sine equals opposite over hypotenuse," CAH for "cosine equals adjacent over hypotenuse," and TOA for "tangent equals opposite over adjacent."

Unit Circle Definitions

The unit circle provides a geometric foundation for defining the trigonometric functions sine and cosine for any real number θ, extending beyond the limitations of right triangles to encompass all angles, including those greater than 90° or negative. Consider a circle of radius 1 centered at the origin (0,0) in the Cartesian plane. An angle θ is formed by rotating a ray from the positive x-axis counterclockwise (positive θ) or clockwise (negative θ) to a terminal side that intersects the unit circle at a point P = (x, y). The cosine of θ is defined as the x-coordinate of P, so cos θ = x, and the sine of θ is the y-coordinate, so sin θ = y. This definition ensures that sin²θ + cos²θ = 1 for all θ, as (x, y) lies on the circle x² + y² = 1. The signs of sin θ and cos θ depend on the quadrant in which the terminal side of θ lies. In quadrant I (0 < θ < π/2), both sin θ and cos θ are positive. In quadrant II (π/2 < θ < π), sin θ is positive while cos θ is negative. In quadrant III (π < θ < 3π/2), both are negative. In quadrant IV (3π/2 < θ < 2π), sin θ is negative while cos θ is positive. For angles beyond one full rotation or negative values, the position repeats periodically every 2π radians due to the circular nature of the definitions. To evaluate sin θ and cos θ for angles in quadrants II, III, or IV, the reference angle is used, defined as the acute angle between the terminal side of θ and the nearest x-axis. The reference angle θ' equals θ for quadrant I, π - θ for quadrant II, θ - π for quadrant III, and 2π - θ for quadrant IV (adjusting for coterminal angles if necessary). The values are then computed as sin θ = ± sin θ' and cos θ = ± cos θ', with the sign determined by the quadrant. This approach leverages known values from quadrant I while accounting for the geometric positions on the unit circle. When angles are measured in radians, θ represents the arc length along the unit circle from the positive x-axis to point P, since the radius is 1 and arc length s = rθ simplifies to s = θ. This radian measure facilitates natural connections between angles, arc lengths, and trigonometric functions, as the coordinates (cos θ, sin θ) directly correspond to positions traversed by that arc. The unit circle definitions thus generalize the ratios from right triangles—where sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse for acute angles—to arbitrary θ by embedding the triangle within the circle's geometry.

Exact Values

Common Algebraic Values

The exact values of trigonometric functions at standard angles such as 0°, 30°, 45°, 60°, and 90° (or their radian equivalents 0, π/6, π/4, π/3, and π/2) are derived primarily from special right triangles, where the side lengths follow specific ratios that allow algebraic expressions without approximation. For the 45° angle, consider an isosceles right triangle with legs of length 1 and hypotenuse √2, formed by placing the right angle at the origin and the equal angles at 45°. The sine of 45° is the opposite side over the hypotenuse, yielding sin(45°) = 1/√2 = √2/2; similarly, cos(45°) = adjacent/hypotenuse = √2/2, and tan(45°) = opposite/adjacent = 1. The 30° and 60° angles arise from a 30°-60°-90° triangle, obtained by bisecting an with side length 2, resulting in side ratios of 1 (opposite 30°), √3 (opposite 60°), and 2 (hypotenuse). Thus, sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 = √3/3; for 60°, sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3. At 0° and 90°, the values follow directly from the right triangle definitions or the unit circle, where sin(0°) = 0, cos(0°) = 1, tan(0°) = 0, sin(90°) = 1, cos(90°) = 0, and tan(90°) is undefined due to division by zero. For angles at multiples of π/2 beyond the first quadrant, such as π/2, π, 3π/2, and 2π (equivalent to 90°, 180°, 270°, and 360°), the unit circle provides the coordinates of intersection points, with cos(θ) as the x-coordinate and sin(θ) as the y-coordinate on the circle of radius 1. These yield sin(π/2) = 1, cos(π/2) = 0, tan(π/2) undefined; sin(π) = 0, cos(π) = -1, tan(π) = 0; sin(3π/2) = -1, cos(3π/2) = 0, tan(3π/2) undefined; sin(2π) = 0, cos(2π) = 1, tan(2π) = 0. The following table summarizes the exact values for sine, cosine, and tangent at these standard angles:
Angle (degrees)Angle (radians)sincostan
0°0010
30°π/61/2√3/2√3/3
45°π/4√2/2√2/21
60°π/3√3/21/2√3
90°π/210undefined
180°π0-10
270°3π/2-10undefined
360°2π010
These values are foundational for computations in trigonometry and arise consistently from the geometric definitions.

Properties and Identities

Periodicity and Symmetry

Trigonometric functions exhibit periodicity, meaning their values repeat at regular intervals. The sine and cosine functions are periodic with a fundamental period of $2\pi, such that \sin(\theta + 2\pi) = \sin \theta and \cos(\theta + 2\pi) = \cos \theta for all \theta. This periodicity arises from their definitions on the , where angles differing by $2\pi correspond to the same point. The tangent and cotangent functions have a smaller fundamental period of \pi, with \tan(\theta + \pi) = \tan \theta and \cot(\theta + \pi) = \cot \theta. These periods reflect the repeating patterns in their ratios of sine and cosine. The graphs of these functions illustrate their periodicity over one full period. For the sine function, the graph over [0, 2\pi] starts at (0, 0), rises to a maximum of 1 at \pi/2, returns to 0 at \pi, reaches a minimum of -1 at $3\pi/2, and ends at 0 at $2\pi, forming a smooth wave that repeats thereafter. The cosine graph over the same interval begins at (0, 1), decreases to 0 at \pi/2, to -1 at \pi, back to 0 at $3\pi/2, and to 1 at $2\pi, shifted horizontally by \pi/2 from the sine wave. For tangent, one period from -\pi/2 to \pi/2 (excluding endpoints) shows a curve passing through (0, 0), approaching +\infty as \theta nears \pi/2 from the left, and -\infty from the right of -\pi/2, with vertical asymptotes at odd multiples of \pi/2. The cotangent graph over (0, \pi) starts near +\infty just after 0, passes through (\pi/2, 0), and approaches -\infty near \pi, with vertical asymptotes at integer multiples of \pi. Symmetry properties further characterize these functions, classifying them as even or odd based on their behavior under negation of the argument. The cosine function is even, satisfying \cos(-\theta) = \cos \theta, which implies its graph is symmetric about the y-axis; for example, the value at \theta matches that at -\theta. In contrast, the sine, tangent, and cotangent functions are odd, with \sin(-\theta) = -\sin \theta, \tan(-\theta) = -\tan \theta, and \cot(-\theta) = -\cot \theta; their graphs exhibit rotational symmetry about the origin, such that the point (\theta, f(\theta)) maps to (-\theta, -f(\theta)). These symmetries are evident in the graphs: the cosine wave mirrors itself across the y-axis, while the sine wave rotates 180 degrees around the origin to coincide with itself, and similar point symmetry holds for tangent and cotangent within their periodic branches between asymptotes.

Pythagorean Identities

The Pythagorean identities form a cornerstone of trigonometric theory, expressing relationships between the squares of sine and cosine, and their reciprocals. The fundamental identity is \sin^2 \theta + \cos^2 \theta = 1, which holds for all real angles θ. This identity derives directly from the unit circle definition of the trigonometric functions. On the unit circle centered at the origin with radius 1, any point corresponding to angle θ has coordinates (\cos \theta, \sin \theta). The equation of the circle, x^2 + y^2 = 1, substitutes to yield \cos^2 \theta + \sin^2 \theta = 1. The identity extends to the other pairs of cofunctions by algebraic manipulation. Dividing the primary identity by \cos^2 \theta (where \cos \theta \neq 0) gives \tan^2 \theta + 1 = \sec^2 \theta. Similarly, dividing by \sin^2 \theta (where \sin \theta \neq 0) yields $1 + \cot^2 \theta = \csc^2 \theta. These forms connect the tangent-secant and cotangent-cosecant pairs, respectively. In practice, the Pythagorean identities simplify trigonometric expressions and aid in solving equations. For example, they allow rewriting \sin^2 \theta as $1 - \cos^2 \theta to express one function in terms of another, which is useful in integration or verifying equalities. Another application involves simplifying ratios like \frac{\sin^2 \theta}{\cos^2 \theta} + 1, which reduces to \sec^2 \theta using the tangent-secant form.

Sum and Difference Formulas

The sum and difference formulas for the sine and cosine functions express the trigonometric values of summed or differenced angles in terms of products of the individual functions. These identities are fundamental in trigonometry and originate from geometric considerations in the second century CE, as developed by Ptolemy in his Almagest using chord lengths in a circle. The addition formula for sine is given by \sin(A + B) = \sin A \cos B + \cos A \sin B, and the subtraction formula by \sin(A - B) = \sin A \cos B - \cos A \sin B. Similarly, for cosine, the addition formula is \cos(A + B) = \cos A \cos B - \sin A \sin B, and the subtraction formula is \cos(A - B) = \cos A \cos B + \sin A \sin B. These formulas can be derived using the geometry of the unit circle. Consider points on the unit circle centered at the origin: let Q = (\cos A, \sin A) and S = (\cos(-B), \sin(-B)) = (\cos B, -\sin B). The distance between Q and S is equated to the distance between the point at angle A + B, R = (\cos(A + B), \sin(A + B)), and the origin point P = (1, 0). Applying the distance formula yields PR^2 = 2 - 2\cos(A + B) and QS^2 = 2 - 2(\cos A \cos B - \sin A \sin B), leading directly to the cosine addition formula; the sine formulas follow by considering rotations or auxiliary angles. An alternative geometric derivation employs Ptolemy's theorem for a cyclic quadrilateral inscribed in a unit circle with diameter BC = 1. For angles A and B, the sides and diagonals correspond to sines and cosines via the law of sines. Applying Ptolemy's theorem—stating that the product of the diagonals equals the sum of the products of opposite sides—reformulates to \sin(A + B) = \sin A \cos B + \cos A \sin B. The subtraction formula arises similarly by adjusting the configuration with BC as the diameter. This approach, rooted in Euclidean geometry (Elements, Book III, Propositions 20 and 21), underscores the formulas' ancient origins. The tangent addition and subtraction formulas are derived from the sine and cosine identities by division: \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}, \quad \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}, provided the denominators are nonzero. These follow immediately from \tan \theta = \sin \theta / \cos \theta and simplifying the quotients of the sum and difference formulas. A notable special case occurs when A and B are complementary angles, satisfying A + B = \pi/2. Substituting into the sine addition formula yields \sin(\pi/2) = 1 = \sin A \cos B + \cos A \sin B, which, with B = \pi/2 - A, simplifies to \sin^2 A + \cos^2 A = 1, confirming the Pythagorean identity.

Multiple-Angle Formulas

Multiple-angle formulas express the trigonometric functions of multiple angles, such as 2θ, 3θ, or θ/2, in terms of functions of θ alone. These formulas are derived from the sum and difference formulas by substituting specific angles, enabling simplification of expressions involving repeated angles.

Double-Angle Formulas

The double-angle formulas arise by applying the sum formulas with both angles equal to θ. For sine, \sin(2\theta) = \sin(\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta = 2 \sin \theta \cos \theta. This identity is fundamental for expanding products of sine and cosine. For cosine, the derivation yields multiple equivalent forms: \cos(2\theta) = \cos(\theta + \theta) = \cos \theta \cos \theta - \sin \theta \sin \theta = \cos^2 \theta - \sin^2 \theta. Using the Pythagorean identity \cos^2 \theta + \sin^2 \theta = 1, this can be rewritten as \cos(2\theta) = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta. The form $2\cos^2 \theta - 1 serves as a power-reduction formula, expressing \cos^2 \theta as \frac{1 + \cos(2\theta)}{2}, which reduces the power of the cosine in integrals or series expansions. Similarly, \sin^2 \theta = \frac{1 - \cos(2\theta)}{2}. These power-reduction variants facilitate averaging over angles or simplifying polynomial expressions in trigonometry.

Triple-Angle Formulas

Triple-angle formulas extend the double-angle results by adding another θ. For sine, \sin(3\theta) = \sin(2\theta + \theta) = \sin(2\theta) \cos \theta + \cos(2\theta) \sin \theta = 2 \sin \theta \cos \theta \cos \theta + (\cos^2 \theta - \sin^2 \theta) \sin \theta. Simplifying further using \cos^2 \theta = 1 - \sin^2 \theta gives \sin(3\theta) = 3 \sin \theta - 4 \sin^3 \theta. This cubic form is particularly useful in solving equations like the triple-angle equation for sine, which relates to the geometry of trisecting angles. A similar derivation for cosine yields \cos(3\theta) = 4 \cos^3 \theta - 3 \cos \theta, though the focus here is on the sine variant as a direct extension.

Half-Angle Formulas

Half-angle formulas solve for functions of θ/2 in terms of θ, derived by treating the double-angle formulas as equations in the half-angle. For sine, starting from \cos \theta = 1 - 2 \sin^2(\theta/2), rearranging gives \sin^2(\theta/2) = \frac{1 - \cos \theta}{2}, so \sin(\theta/2) = \pm \sqrt{\frac{1 - \cos \theta}{2}}. The sign depends on the quadrant of θ/2. This formula is essential for integrating expressions like \int \sqrt{1 - \cos \theta} \, d\theta or computing exact values for angles like 22.5°. Analogously, from \cos \theta = 2 \cos^2(\theta/2) - 1, \cos(\theta/2) = \pm \sqrt{\frac{1 + \cos \theta}{2}}. These identities, along with the tangent half-angle formula \tan(\theta/2) = \frac{\sin \theta}{1 + \cos \theta}, support substitutions in rational functions of sine and cosine.

Differentiation and Integration Formulas

The derivatives of the basic trigonometric functions are fundamental results in calculus, assuming the argument is in radians. The derivative of the sine function is given by \frac{d}{dx} \sin x = \cos x, while the derivative of the cosine function is \frac{d}{dx} \cos x = -\sin x. These formulas can be derived using the limit definition of the derivative, relying on the standard limits \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 and \lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = 0. For the tangent function, defined as \tan x = \frac{\sin x}{\cos x}, the derivative follows from the quotient rule: \frac{d}{dx} \tan x = \sec^2 x. This is obtained by differentiating the numerator and denominator separately and applying the quotient rule formula \left( \frac{u}{v} \right)' = \frac{u' v - u v'}{v^2}, yielding \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x. Higher-order derivatives of \sin x and \cos x exhibit a cyclic pattern with period four. Specifically, the first derivative of \sin x is \cos x, the second is -\sin x, the third is -\cos x, and the fourth is \sin x, after which the pattern repeats. Similarly, for \cos x, the derivatives cycle as -\sin x, -\cos x, \sin x, and \cos x. This periodicity arises directly from repeated application of the basic derivative rules and is useful for computing nth-order derivatives without exhaustive differentiation. The indefinite integrals of the basic trigonometric functions are the antiderivatives, which reverse the differentiation process. Thus, \int \sin x \, dx = -\cos x + C, and \int \cos x \, dx = \sin x + C. For the tangent function, the integral is derived using the substitution u = \cos x, so du = -\sin x \, dx and \tan x = \frac{\sin x}{\cos x}, leading to \int \tan x \, dx = \int \frac{\sin x}{\cos x} \, dx = -\int \frac{du}{u} = -\ln |u| + C = -\ln |\cos x| + C, which is equivalent to \ln |\sec x| + C. For integrals of powers of sine, such as \int \sin^n x \, dx where n > 1 is an integer, a reduction formula allows recursive computation by lowering the power. The formula is \int \sin^n x \, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2} x \, dx. This is proved using : let u = \sin^{n-1} x and dv = \sin x \, dx, so du = (n-1) \sin^{n-2} x \cos x \, dx and v = -\cos x. Then, \int u \, dv = uv - \int v \, du yields the relation, which can be rearranged to the reduction form. Similar formulas exist for powers of cosine and other trigonometric functions, facilitating evaluation of higher-power integrals.

Advanced Definitions

Power Series and Infinite Products

The power series expansions, also known as Maclaurin series, provide analytic definitions of the functions that are valid for all real numbers. These series arise from the repeated of the functions and evaluation at zero, reflecting their , entire nature in the restricted to real arguments. The convergence radius is infinite, ensuring representation everywhere on the real line without singularities. The for sine is given by \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots, which converges absolutely for all real x. Similarly, the series for cosine is \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots, also converging for all real x. These expansions encode the periodic and oscillatory behavior through alternating signs and factorial denominators that grow rapidly, enabling efficient computation for small to moderate |x|. These series can be derived systematically by solving the second-order y'' + y = 0, which both satisfy. Assume a solution y(x) = \sum_{n=0}^{\infty} a_n x^n. Differentiating twice and substituting yields the recurrence a_{n+2} = -a_n / ((n+1)(n+2)) for n \geq 0. For , the initial conditions y(0) = 0 and y'(0) = 1 fix a_0 = 0, a_1 = 1, producing only odd-powered terms. For , y(0) = 1 and y'(0) = 0 yield a_0 = 1, a_1 = 0, resulting in even powers. This method confirms the coefficients without relying on geometric definitions. An alternative perspective on the series emerges from basic , where serve as basis functions for expansions of periodic functions, and their own series align with the properties over intervals like [-\pi, \pi]. The sine function also possesses an elegant representation, first derived by Leonhard Euler in 1748 by analogy to the . Considering \sin x / x as an "infinite-degree polynomial" with zeros at x = \pm n\pi for positive integers n, Euler expressed it as \frac{\sin x}{x} = \prod_{n=1}^{\infty} \left(1 - \frac{x^2}{n^2 \pi^2}\right), or equivalently, \sin x = x \prod_{n=1}^{\infty} \left(1 - \frac{x^2}{n^2 \pi^2}\right). This product converges uniformly on compact subsets of the real line excluding the zeros at integer multiples of \pi. The derivation involves comparing the Taylor series coefficients of the product to those of \sin x / x, leveraging the known roots and leading coefficient to match the expansions term by term. This form highlights the function's zeros and connects to broader analytic number theory, such as evaluations of the Riemann zeta function at even integers.

Exponential and Complex Definitions

One of the most elegant ways to define the trigonometric functions analytically is through their relationship with the , particularly via , which states that for a \theta, e^{i\theta} = \cos \theta + i \sin \theta. This formula links the with purely imaginary exponents to the trigonometric functions on the unit circle in the . From this, the trigonometric functions can be expressed explicitly as \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}, \quad \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}. These definitions provide a unified view of sine and cosine as the imaginary and real parts, respectively, of the complex exponential e^{i\theta}. The exponential definitions extend naturally to complex arguments, allowing trigonometric functions to be defined for any complex number z. Thus, the complex sine and cosine are given by \sin z = \frac{e^{iz} - e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz} + e^{-iz}}{2}. These functions are entire, meaning they are holomorphic everywhere in the complex plane, and they satisfy many of the same identities as their real counterparts, though with periodic strips rather than full periodicity. A notable connection arises between these complex trigonometric functions and the hyperbolic functions, where the hyperbolic sine relates to the complex sine via \sinh x = -i \sin(ix) for real x. This relation highlights the analogy between trigonometric and hyperbolic functions through imaginary arguments.

Differential and Integral Definitions

Trigonometric functions can be defined as the solutions to the second-order y'' + y = 0. The cosine function is the unique solution satisfying the initial conditions \cos 0 = 1 and \cos' 0 = 0, while the sine function is the unique solution satisfying \sin 0 = 0 and \sin' 0 = 1. The general solution to this equation is a y(x) = A \cos x + B \sin x, where the constants A and B are determined by the initial conditions. The uniqueness of these solutions follows from the existence and uniqueness theorem for linear ordinary differential equations with continuous coefficients, which applies here since the coefficients are constants. For the second-order equation, it can be rewritten as a first-order system \begin{cases} y' = z \\ z' = -y \end{cases}, to which the applies, guaranteeing a unique solution on the real line given the condition on the right-hand side. An alternative definition uses integrals in a mutually recursive manner: \sin x = \int_0^x \cos t \, dt and \cos x = 1 - \int_0^x \sin t \, dt. These definitions are consistent because differentiating them yields \cos x = \frac{d}{dx} \sin x and -\sin x = \frac{d}{dx} \cos x, which imply that both functions satisfy the y'' + y = 0 with the appropriate initial conditions. This differential and integral approach parallels the definition of functions as solutions to y' = y, with emerging as the real and imaginary parts of the complex exponential e^{ix}. Complex extensions of these definitions are explored further in exponential formulations.

Functional Equation Definitions

Trigonometric functions can be axiomatized through functional equations that capture their additive properties, independent of geometric or analytic definitions. Consider real-valued functions S and C defined on \mathbb{R} satisfying the Pythagorean identity S(x)^2 + C(x)^2 = 1 for all x \in \mathbb{R}, along with initial conditions S(0) = 0 and C(0) = 1, and the additive relations C(x + y) = C(x)C(y) - S(x)S(y), \quad S(x + y) = S(x)C(y) + C(x)S(y) for all x, y \in \mathbb{R}. These equations resemble a Cauchy-type addition formula tailored to the unit circle constraint. If S or C is continuous at a single point, then there exists a constant a \neq 0 such that C(x) = \cos(ax) and S(x) = \sin(ax) for all x \in \mathbb{R}. The additive formula for sine specifically, S(x + y) = S(x)C(y) + C(x)S(y), extends the classical Cauchy functional equation f(x + y) = f(x) + f(y) by incorporating the auxiliary function C, reflecting the interdependent nature of . Without continuity, pathological solutions exist using Hamel bases, but regularity conditions like , monotonicity, or measurability guarantee uniqueness up to the scaling parameter a. Analyticity further ensures the solutions are entire functions, aligning with their periodic behavior. For cosine alone, d'Alembert's provides a standalone : a f: \mathbb{R} \to \mathbb{R} satisfies f(x + y) + f(x - y) = 2f(x)f(y) for all x, y \in \mathbb{R}, with f(0) = 1. Under , the solutions are f(x) = \cos(ax) for some a \in \mathbb{R}. This equation originates from studies in wave propagation and has been generalized to groups, where continuous solutions on \mathbb{R} remain trigonometric. Boundedness or periodicity assumptions also yield the same uniqueness. These functional equations connect to , as the pair (C(x), S(x)) represents a continuous from the additive group (\mathbb{R}, +) to the SO(2), the group of 2D rotations, via the matrix \begin{pmatrix} C(x) & -S(x) \\ S(x) & C(x) \end{pmatrix}. The is $2\pi \mathbb{Z}/|a|, ensuring the covering map property. This perspective unifies the algebraic relations with the geometric interpretation of rotations.

Inverse Functions

Definitions and Principal Values

The inverse trigonometric functions, also known as arcus functions, provide the angles whose trigonometric ratios equal a given value, reversing the action of the standard trigonometric functions. Due to the periodic nature of trigonometric functions, which results in multiple angles yielding the same ratio, these inverses are defined with specific restrictions on their ranges to ensure they are single-valued and bijective, producing a unique principal value for each input in their domain. The arcsine function, denoted \arcsin(y), is defined as the angle \theta such that \sin \theta = y and -\pi/2 \leq \theta \leq \pi/2. Its domain is the closed interval [-1, 1], corresponding to the range of the sine function over its principal interval. Similarly, the arccosine function, \arccos(y), satisfies \cos \theta = y with $0 \leq \theta \leq \pi, also defined on [-1, 1]. The arctangent function, \arctan(y), is the angle \theta where \tan \theta = y and -\pi/2 < \theta < \pi/2, with a domain of all real numbers \mathbb{R}, reflecting the unbounded range of the tangent function in its principal interval. These restrictions ensure bijectivity: the sine function is one-to-one and onto from [-\pi/2, \pi/2] to [-1, 1], the cosine from [0, \pi] to [-1, 1], and the tangent from (-\pi/2, \pi/2) to \mathbb{R}. The graph of \arcsin(y) is an increasing S-shaped curve, symmetric about the origin, starting at ( -1, -\pi/2 ) and ending at ( 1, \pi/2 ). The graph of \arccos(y) decreases from ( -1, \pi ) to ( 1, 0 ), while \arctan(y) approaches -\pi/2 as y \to -\infty and \pi/2 as y \to \infty, passing through the origin. Reciprocal inverse functions are defined analogously using the reciprocals of the trigonometric functions. The arcsecant, \arcsec(y), is the angle \theta such that \sec \theta = y (or \cos \theta = 1/y) with $0 \leq \theta < \pi/2 or \pi/2 < \theta \leq \pi, defined for y \leq -1 or y \geq 1. The arccosecant, \arccsc(y), satisfies \csc \theta = y (or \sin \theta = 1/y) with -\pi/2 \leq \theta < 0 or $0 < \theta \leq \pi/2, also on y \leq -1 or y \geq 1. The arccotangent, \arccot(y), is the angle \theta where \cot \theta = y (or \tan \theta = 1/y) with $0 < \theta < \pi, defined for all real y. These ranges make the functions bijective over their respective domains, with graphs that are decreasing for \arcsec and \arccsc (except at discontinuities) and decreasing for \arccot.

Properties and Identities

The inverse sine function, arcsin, is an odd function, satisfying arcsin(-x) = -arcsin(x) for all x in the domain [-1, 1]. This property follows from the oddness of the sine function and the principal branch definition of the inverse. The derivatives of the principal inverse trigonometric functions are fundamental for calculus applications. The derivative of arcsin x is given by \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1. This result is obtained via implicit differentiation from the relation sin(y) = x, where y = arcsin x. Similarly, the derivative of arctan x is \frac{d}{dx} \arctan x = \frac{1}{1 + x^2}, \quad x \in \mathbb{R}. This follows from implicit differentiation of tan(y) = x, yielding a secant expression that simplifies using the Pythagorean identity. Key identities include the complementary property for arctangent: arctan x + arctan(1/x) = π/2 for x > 0. For x < 0, the sum is -π/2, reflecting the odd nature of arctan. More generally, the addition formula for arctangent states that \arctan a + \arctan b = \arctan\left( \frac{a + b}{1 - ab} \right) when ab < 1, with adjustments by ±π when ab > 1 to account for the principal range (-π/2, π/2). This formula derives from the tangent addition rule applied to the whose tangents are a and b. Indefinite integrals involving these functions often yield inverse trigonometric expressions. For instance, \int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C, \quad |x| < 1. This antiderivative is verified by differentiation, confirming its consistency with the arcsin derivative. Similarly, \int \frac{dx}{1 + x^2} = \arctan x + C, \quad x \in \mathbb{R}, arising directly from the arctan derivative. These integrals are essential in evaluating definite forms and solving differential equations.

Applications

Geometry and Triangles

Trigonometric functions are fundamentally applied in geometry to determine unknown angles and side lengths in right triangles, where the sine, cosine, and tangent ratios relate the sides opposite, adjacent to, and opposite over adjacent to a given acute angle, respectively. In a right triangle with acute angle θ, opposite side a, adjacent side b, and hypotenuse c, these are defined as sin θ = a/c, cos θ = b/c, and tan θ = a/b. To solve for missing elements, one can use these ratios inversely: for example, if the hypotenuse and an acute angle are known, the opposite side is found via a = c sin θ. Such applications allow computation of all sides and angles when at least one side and one acute angle (or two sides) are given, leveraging the Pythagorean theorem for verification. For non-right triangles, the extends these ratios to relate all three sides and opposite : a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius. This equality holds for any and enables solving for unknown or sides when two and any side ( or ) or two sides and an angle opposite one of them () are known. However, the SSA case can be ambiguous, potentially yielding zero, one, or two triangles depending on the height relative to the given side and angle; specifically, if the given angle is acute and the opposite side is shorter than the adjacent side but longer than the adjacent side times the sine of the angle, two triangles may exist. The generalizes the to any triangle, providing c² = a² + b² - 2ab cos C for the side opposite angle C, and cyclic permutations for the others. This formula is particularly useful for solving triangles when two sides and the included angle () or all three sides () are known, allowing computation of the opposite angle via the inverse cosine. It directly relates side lengths to the cosine of the included angle, facilitating solutions where the alone is insufficient. A complementary relation, the , connects differences and sums of sides to tangents of half-: (a - b) / (a + b) = tan((A - B)/2) / tan((A + B)/2). This identity, derived from the laws of and cosines, aids in solving for when sides are known, especially in cases involving half-angle computations, though it is less commonly used than the primary laws. Trigonometric functions also provide a direct formula for the area of any : (1/2)ab sin C, where a and b are two sides and C is the included angle. This expression, equivalent to (1/2)bc sin A or (1/2)ac sin B, derives from the height interpretation of the sine and is applicable even without knowing the third side, making it valuable for geometric computations.

Calculus and Periodic Functions

Trigonometric functions play a central role in the analysis of periodic phenomena through , which allow the representation of any sufficiently smooth as an infinite sum of terms. This decomposition exploits the of these functions over one period, enabling the approximation of complex waveforms by superpositions of simpler harmonic components. The general form is f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)), where the coefficients a_n and b_n are determined by integrals involving the function and the basis functions. This method, introduced by in his seminal work on heat conduction, revolutionized the study of in . In differential equations, trigonometric functions provide the natural solutions to second-order linear homogeneous equations modeling oscillatory behavior, such as the simple . The equation \frac{d^2x}{dt^2} + \omega^2 x = 0 has the general solution x(t) = A \cos(\omega t + \phi), where A is the and \phi is the shift, determined by initial conditions. This form arises from the r^2 + \omega^2 = 0, yielding complex roots r = \pm i\omega, whose real and imaginary parts correspond to cosine and sine functions via . Such solutions highlight the periodic nature inherent in undamped oscillatory systems. Trigonometric substitution is a powerful integration technique for handling integrals involving square roots of quadratic expressions, particularly those resembling trigonometric identities. For the integral \int \frac{dx}{\sqrt{x^2 + a^2}}, the substitution x = a \tan \theta simplifies the expression, as \sqrt{x^2 + a^2} = a \sec \theta and dx = a \sec^2 \theta \, d\theta, leading to \int \sec \theta \, d\theta = \ln |\sec \theta + \tan \theta| + C. Back-substituting yields \sinh^{-1}(x/a) + C or equivalently \ln |x + \sqrt{x^2 + a^2}| + C, demonstrating how trigonometric functions facilitate the evaluation of otherwise intractable integrals. For small-angle approximations, the Taylor series expansions of provide essential simplifications in applications. The series for \sin x is x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots, so for small x, \sin x \approx x; similarly, \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \approx 1 - \frac{x^2}{2}. These linear and approximations are derived by evaluating the and its at zero, offering high accuracy for angles near zero radians and aiding in numerical computations and asymptotic analyses.

Physics and Engineering

Trigonometric functions play a pivotal role in modeling dynamic systems across physics and , where they describe periodic behaviors, resolve forces, and analyze signals in real-world applications. In physics, functions capture the oscillatory nature of phenomena like , while in engineering, they enable precise calculations for circuits and mechanical systems. These applications leverage the periodic properties of trigonometric functions to represent time-varying quantities accurately. In waves and oscillations, trigonometric functions model sinusoidal variations essential for understanding sound, , and mechanical vibrations. For instance, the displacement in or a is often expressed as y = A \sin(2\pi f t + \phi), where A is the , f is the , t is time, and \phi is the phase angle; this form describes the periodic up-and-down motion in sound propagating through air or in electromagnetic spectra. Sound waves, for example, are modeled this way to predict pressure variations that the human ear perceives as pitch and volume, while use similar sinusoidal profiles to explain interference patterns in . Trigonometric functions are crucial for resolving vectors and forces in , particularly in , where initial components are decomposed using . The horizontal component is v_x = v \cos \theta, and the vertical component is v_y = v \sin \theta, with v as the launch speed and \theta as the angle relative to the horizontal; this allows engineers to predict trajectories for applications like or launches. In under , these components separate the motion into independent horizontal (constant ) and vertical (accelerated) parts, enabling calculations of and maximum height without solving coupled equations. In signal processing, the Fast Fourier Transform (FFT) relies on trigonometric functions to decompose complex signals into their frequency components, facilitating analysis in fields like audio engineering and telecommunications. The FFT computes the discrete Fourier transform efficiently, expressing a time-domain signal as a sum of sines and cosines at various frequencies, which reveals dominant periodic elements in noisy data. This technique is widely used to filter signals or identify vibrations in machinery, transforming raw sensor data into interpretable spectra for fault detection. Electrical engineering employs trigonometric functions to analyze alternating current (AC) circuits, where phase shifts between voltage and current are quantified using tangent. In a resistive-inductive (RL) circuit, the phase angle \phi satisfies \tan \phi = X_L / R, with X_L as inductive reactance and R as resistance; this determines power factor and efficiency in power distribution systems. Similarly, in RC circuits, \tan \phi = -X_C / R accounts for capacitive effects, guiding the design of filters and oscillators in electronics.

Historical and Linguistic Aspects

Historical Development

The origins of trigonometric functions trace back to ancient astronomy and , where early mathematicians developed tables to solve practical problems in and measurement. In the 2nd century BCE, the Greek astronomer compiled the first known tables of , which measured the length of a subtending an in a of fixed , laying the foundational work for what would become . These tables, though not surviving intact, influenced subsequent Hellenistic and later traditions by providing a systematic approach to angular calculations. The Alexandrian astronomer (c. 100–170 CE) advanced this by establishing key identities, such as the in trigonometric form (sin²θ + cos²θ = 1, expressed in chords), and providing an equivalent of the : a / sin A = b / sin B = c / sin C. Around 500 CE, the Indian mathematician advanced this further by introducing the sine function, denoted as "jya" (meaning ), and computing sine tables for astronomical purposes in his text , marking the first explicit use of sine as a distinct trigonometric . During the (8th–13th centuries CE), scholars refined and expanded trigonometric methods, integrating Greek and Indian knowledge with original innovations driven by advancements in astronomy. (c. 858–929 CE), a prominent astronomer from , produced highly accurate sine tables and was the first to express trigonometric functions as lengths rather than proportions, while also contributing to by refining Ptolemaic models for planetary motion. In the 9th–10th centuries, Islamic mathematicians introduced the tangent function, initially conceptualized as the "extended shadow" (the shadow cast by a horizontal rod on a vertical wall), which employed in his calculations; this was later formalized by (973–1048 CE) alongside the cotangent. Abu al-Wafa (940–998 CE) further derived addition formulas, including the double-angle identity sin(2x) = 2 sin(x) cos(x). The European Renaissance (15th–16th centuries) saw the revival and systematization of trigonometry as a standalone mathematical discipline, spurred by printing technology and astronomical needs. (Johannes Müller, 1436–1476) authored De triangulis omnimodis around 1464 (published 1533), the first comprehensive European treatise on trigonometry, which emphasized sines and their inverses in plane and spherical contexts, drawing from Islamic sources like . His student Georg Rheticus (1514–1574) advanced computational aspects in 1542 by publishing extensive tables of sines and cosines to 10 decimal places, defining functions geometrically in terms of right triangles with unit , and assisting in the trigonometric sections of Copernicus's De revolutionibus. In the , Leonhard Euler (1707–1783) propelled trigonometric functions into analytic by treating sine, cosine, and others as standalone functions rather than geometric ratios, introducing key identities and the complex exponential representation in his (1748). In the 19th and early 20th centuries, trigonometric functions gained deeper theoretical foundations through and series expansions. (1768–1830) revolutionized their application in the 1820s by developing , which decompose arbitrary periodic functions into infinite sums of sines and cosines, as detailed in his Théorie analytique de la chaleur (1822) while solving the . (1826–1866) extended this in his 1854 habilitation dissertation by investigating the conditions under which functions could be represented by trigonometric series, linking them to integrability and complex variable theory, thereby bridging with Riemann surfaces and holomorphic functions.

Etymology

The term "sine" originates from the Latin sinus, meaning "a bend, curve, fold in a garment, or bosom," which entered mathematical usage in the mid-12th century through Gherardo of Cremona's translation of geometrical texts. The jiba, denoting the "chord of an arc" or sine, derived from the jya ("bowstring"), but was mistranslated from jaib ("" or "bosom"), leading to the adoption of sinus as a fitting equivalent due to its of a fold or inlet. This linguistic path reflects the transmission of trigonometric concepts from through scholarship to , where the term first appeared in English in the 1590s. The name "cosine," introduced in the 1630s, is a contraction of "co. sinus," abbreviating the Medieval Latin complementi sinus ("sine of the complement"). Here, the prefix "co-" stems from Latin complementum ("that which fills up or completes"), referring to the complementary angle in a , as the cosine of an angle equals the sine of its complement (90° minus the angle). This nomenclature, popularized by English around 1620, underscores the relational symmetry between in early trigonometric tables and calculations. "Tangent" derives from the Latin tangentem (nominative tangens), the present participle of tangere ("to touch"), introduced as a trigonometric term in 1583 by Danish Thomas Fincke in his work Geometria Rotundi. The name evokes the geometric construction where the represents the length of a line from the origin touching the unit circle at one point, emphasizing contact without crossing. Similarly, "cotangent," a 17th-century contraction of "co. tangent," applies the "co-" to denote the tangent of the complementary , mirroring the structure of cosine. The terms "secant" and "cosecant" trace to Latin secantem (nominative secans), from secare ("to cut"), also coined by Fincke in 1583 to describe a line intersecting a at two points. "Cosecant," analogously, combines the "co-" with secant, signifying the secant of the complementary . "Cotangent," as noted, follows the same complementary pattern with . These names, rooted in , highlight the functions' roles as reciprocals of cosine and sine, respectively, in extending basic ratios beyond the unit .