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Secant

The secant function, denoted \sec \theta, is one of the six fundamental trigonometric functions and is defined as the reciprocal of the cosine function: \sec \theta = \frac{1}{\cos \theta}, where \cos \theta \neq 0. In the context of a right triangle with an acute angle \theta, the secant represents the ratio of the hypotenuse length to the length of the adjacent side. The term "secant," derived from the Latin secāre meaning "to cut," was first introduced in its trigonometric sense by Danish mathematician Thomas Fincke in his 1583 treatise Geometriae rotundi. The secant function is periodic with a period of $2\pi, matching that of the cosine, and its graph consists of branches separated by vertical asymptotes at \theta = \frac{\pi}{2} + k\pi for any integer k, where the cosine is zero. Its range is (-\infty, -1] \cup [1, \infty), reflecting that its absolute value is always at least 1, and it is undefined at points where the cosine vanishes. Key identities involving secant include \sec^2 \theta = 1 + \tan^2 \theta, derived from the Pythagorean theorem in trigonometric form, and \sec(-\theta) = \sec \theta, indicating even symmetry. Beyond its role in right-triangle trigonometry and unit-circle definitions, the secant function appears in calculus for integrals and derivatives—such as \frac{d}{d\theta} \sec \theta = \sec \theta \tan \theta—and in applications like optics, physics, and engineering for modeling periodic phenomena. The inverse secant, or arcsecant, \sec^{-1} y, is defined for y \in (-\infty, -1] \cup [1, \infty) with range [0, \pi] \setminus \{\frac{\pi}{2}\}. While primarily a trigonometric concept, "secant" also denotes a line intersecting a curve at two points in geometry, serving as a precursor to the tangent in limit processes for derivatives.

Geometry

Secant line

In geometry, a secant line is defined as a straight line that intersects a curve at two or more distinct points. This contrasts with a tangent line, which intersects the curve at precisely one point, and with lines that do not intersect the curve at any points. For a circle, a secant line specifically intersects the circumference at exactly two points, forming a chord between them, whereas lines external to the circle pass without intersection. The term "secant" derives from the Latin secare, meaning "to cut," reflecting the line's intersection with the curve. To construct a secant line, select any two distinct points on the curve and draw the unique straight line passing through them; this line extends infinitely in both directions and intersects the curve at those points (and potentially others for higher-degree curves). For example, consider a circle centered at the origin with radius 1: a secant line might pass through points (1,0) and (0,1), intersecting the circle at these two locations and creating a chord segment between them (visualize a diagram where the line slices across the circle, marking the entry and exit points). Similarly, for an ellipse stretched along the x-axis, a secant line could intersect at two points symmetric about the major axis, such as near the vertices, forming a chord that highlights the curve's oval shape (imagine a diagram showing the line crossing the elongated boundary twice). In the case of a parabola like y = x², a secant line with a slight positive slope might intersect at two points, one on each side of the vertex, illustrating the curve's openness (depict this in a diagram with the line cutting through the upward-opening arc). Secant lines have foundational applications in approximating the behavior of curves. By choosing two points close together on a curve, the slope of the secant line between them provides an average rate of change that approximates the instantaneous slope (tangent) at a point midway between them, serving as a conceptual precursor to derivatives in calculus. This geometric tool thus bridges basic line constructions with more advanced analytical concepts.

Properties and theorems

One fundamental property of secant lines in circle geometry is encapsulated in the power of a point theorem. For a point P outside a circle, if two secant lines emanate from P and intersect the circle at points A and B on one line and C and D on the other, the product of the lengths of the secant segments from P to the farther intersection points equals the product to the nearer ones: PA \cdot PB = PC \cdot PD. This equality holds regardless of the directions of the secants, as long as they both originate from P and intersect the circle at two points each. The theorem extends to cases involving one secant and one tangent or two intersecting chords inside the circle, maintaining the same power value for the point relative to the circle. Another key theorem concerns the angles formed by secants. The secant-secant angle theorem states that when two secants intersect at a point outside the circle, the measure of the resulting angle is half the difference of the measures of the two intercepted arcs: the far arc between the farther intersection points and the near arc between the nearer ones. m\angle = \frac{1}{2} (m(\text{far arc}) - m(\text{near arc})). This result derives from inscribed angle properties and similarity of triangles formed by the secants and chords. It applies symmetrically to configurations with two tangents or a tangent and a secant from the external point. In the broader context of conic sections, secant lines play a central role in projective geometry through the pole-polar relation. For a conic section, the polar of an external point is the line connecting the contact points of tangents from that point; if a secant through the point intersects the conic at two points, its pole is the intersection of the tangents at those points. This duality interchanges points and lines, preserving incidence and enabling harmonic properties in projective configurations. Such relations facilitate the study of conic envelopes and dual conics without coordinates. A limiting case of the secant line occurs when the two intersection points with the circle coincide, causing the secant to approach the tangent line at that point. This geometric intuition underpins the definition of the tangent as the limiting position of secants, bridging classical geometry to differential calculus where the slope of the tangent represents the instantaneous rate of change.

Trigonometry

Definition

In trigonometry, the secant function, denoted as \sec \theta, is defined as the reciprocal of the cosine function: \sec \theta = \frac{1}{\cos \theta}, where \theta is an angle measured in radians or degrees. This definition originates from the geometric concept of a secant line, which "cuts" a circle at two points, extending to the functional reciprocal in the context of right triangles or the unit circle. The domain of the secant function consists of all real numbers \theta except those where \cos \theta = 0, specifically \theta = \frac{\pi}{2} + k\pi for any integer k. The range is the set (-\infty, -1] \cup [1, \infty), reflecting that secant values are either at least 1 in absolute value or undefined at points of vertical asymptotes. On the unit circle, where a point is given by (\cos \theta, \sin \theta), the secant function corresponds to the reciprocal of the x-coordinate, \sec \theta = \frac{1}{x}, provided x \neq 0. This geometric interpretation aligns with the right-triangle definition, where \sec \theta equals the hypotenuse divided by the adjacent side. Common values of the secant function for standard angles are derived directly from cosine values and are summarized in the following table (angles in degrees and radians, with exact forms):
Angle (degrees)Angle (radians)\sec \theta
001
30\pi/6$2/\sqrt{3} or \frac{2\sqrt{3}}{3}
45\pi/4\sqrt{2}
60\pi/32
90\pi/2undefined
These values illustrate the function's behavior for key angles in the first quadrant.

Identities and relations

The secant function is the reciprocal of the cosine function, expressed as \sec \theta = \frac{1}{\cos \theta}, provided \cos \theta \neq 0. This reciprocal relation directly connects secant to cosine and facilitates derivations of other identities. A key relation to the tangent function follows from the definition of tangent as the ratio of sine to cosine: \tan \theta = \frac{\sin \theta}{\cos \theta} = \sec \theta \cdot \sin \theta, again assuming \cos \theta \neq 0. The Pythagorean identity in terms of secant and tangent is a variant obtained by dividing \sin^2 \theta + \cos^2 \theta = 1 by \cos^2 \theta: \sec^2 \theta - \tan^2 \theta = 1. This identity is fundamental for simplifying expressions involving secant and tangent. For multiple angles, the double-angle formula for secant can be derived from the cosine double-angle identity \cos 2\theta = 2\cos^2 \theta - 1, yielding \sec 2\theta = \frac{\sec^2 \theta}{2 - \sec^2 \theta}, or equivalently from \cos 2\theta = \cos^2 \theta - \sin^2 \theta, \sec 2\theta = \frac{\sec^2 \theta}{1 - \tan^2 \theta}. These forms express secant of double the angle in terms of single-angle secant or tangent values. Using Euler's formula, where \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}, the secant function takes the exponential form \sec \theta = \frac{2}{e^{i\theta} + e^{-i\theta}}. This representation highlights the connection between trigonometric and complex exponential functions without requiring a full proof here. The secant function shares structural similarities with the hyperbolic secant, defined as \sech x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}, analogous to the trigonometric case but with real exponentials replacing the complex ones in the denominator. This analogy arises from the relation \cosh x = \cos(ix), underscoring parallels between hyperbolic and circular functions.

Graph and periodicity

The graph of the secant function, \sec \theta = \frac{1}{\cos \theta}, consists of U-shaped branches defined in the open intervals between its vertical asymptotes, where \cos \theta = 0. These asymptotes occur at \theta = \frac{\pi}{2} + k\pi for any integer k, causing the function to approach positive or negative infinity as \theta nears these points from either side. Between consecutive asymptotes, such as from -\frac{\pi}{2} to \frac{\pi}{2}, the graph forms a symmetric U-shape opening upward, reaching a local minimum of \sec \theta = 1 at \theta = 0, and similarly in other intervals adjusted for sign. The secant function is periodic with a fundamental period of $2\pi, meaning \sec(\theta + 2\pi) = \sec \theta for all \theta in its domain, reflecting the periodicity inherited from the cosine function. It exhibits even symmetry, as \sec(-\theta) = \sec \theta, which contributes to the mirrored branches across the y-axis in its fundamental period. However, due to the placement of asymptotes, the function displays odd-like behavior with respect to shifts by \pi, where \sec(\theta + \pi) = -\sec \theta. Unlike sine or cosine, the secant function has no bounded amplitude, as its range excludes the interval (-1, 1) and extends to \pm \infty near asymptotes; instead, vertical stretches are applied via A \sec \theta, which scales the branches by |A| without altering the infinite extent. Horizontal transformations, such as \sec(k\theta), compress or stretch the period to \frac{2\pi}{|k|}, with k > 0 reducing the period and increasing the frequency of branches and asymptotes. Phase shifts and vertical translations further modify the graph, but the core hyperbolic-like curves between asymptotes remain characteristic. As the reciprocal of cosine, the secant graph inverts and reflects the cosine wave, transforming its smooth oscillations into disjoint hyperbolic branches that bulge outward between the zeros of cosine, which become the asymptotes. Key features include local minima of 1 at \theta = 2k\pi and local maxima of -1 at \theta = (2k+1)\pi (for integer k), corresponding to the extrema of cosine at \pm 1. In intervals where cosine is positive, the branches are U-shaped, decreasing from +\infty to 1 then increasing to +\infty. Where cosine is negative, the branches are inverted U-shaped, increasing from -\infty to -1 then decreasing to -\infty. The positive branches occur where cosine is positive, and negative where cosine is negative, repeating every $2\pi.

Numerical analysis

Secant method

The secant method is an iterative root-finding algorithm in numerical analysis for approximating zeros of a continuous function f(x) = 0. It begins with two initial guesses, x_0 and x_1, assumed to bracket or be near the root, and generates subsequent approximations without requiring derivative evaluations. The method updates the estimate using the recurrence relation x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}, where n \geq 1, provided f(x_n) \neq f(x_{n-1}) to avoid division by zero. This formula arises from solving for the x-intercept of the secant line connecting the points (x_{n-1}, f(x_{n-1})) and (x_n, f(x_n)) on the graph of f. Geometrically, the secant method refines the root approximation by repeatedly drawing a secant line through the two latest points on the function curve and taking its intersection with the x-axis as the next iterate; this process leverages linear interpolation to narrow toward the root. The iteration continues until the change in successive approximations or the function value falls below a specified tolerance. A key advantage of the secant method over the Newton-Raphson method is that it eliminates the need to compute or approximate the derivative f'(x), rendering it applicable to non-differentiable functions or those where differentiation is computationally expensive. Additionally, it requires only one new function evaluation per iteration after the initials, potentially reducing overall computational cost for expensive function evaluations compared to Newton-Raphson's two (function and derivative) per step. To illustrate, consider approximating \sqrt{2} by solving f(x) = x^2 - 2 = 0 with initial guesses x_0 = 1 and x_1 = 2. The first iteration yields x_2 = 2 - \frac{(2)^2 - 2}{(2)^2 - 2 - ((1)^2 - 2)} \cdot (2 - 1) = 2 - \frac{2 \cdot 1}{3} = \frac{4}{3} \approx 1.3333, where f(1) = -1 and f(2) = 2. The second iteration, using x_1 = 4/3 and x_0 = 2, gives f(4/3) \approx -0.2222, so x_3 = \frac{4}{3} - \frac{-0.2222 \cdot (4/3 - 2)}{-0.2222 - 2} \approx 1.4000. Subsequent iterations converge rapidly to \sqrt{2} \approx 1.4142. This example demonstrates the method's efficiency for simple nonlinear equations. The algorithm can be implemented as follows in pseudocode: 
function secant(f, x0, x1, tol, maxiter):
    if abs(f(x1)) < tol:
        return x1
    for n = 1 to maxiter:
        denom = f(x1) - f(x0)
        if abs(denom) < tol:  # Avoid division by near-zero
            break
        x2 = x1 - f(x1) * (x1 - x0) / denom
        if abs(x2 - x1) < tol or abs(f(x2)) < tol:
            return x2
        x0 = x1
        x1 = x2
    return x1  # Or raise error for non-convergence
This outline ensures termination criteria based on approximation difference or function value, with safeguards for numerical stability.

Convergence properties

The secant method demonstrates superlinear convergence with an order of \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618, the golden ratio, under the assumption that the root is simple (i.e., f'(r) \neq 0) and the function is twice continuously differentiable in a neighborhood of the root. This order arises from the asymptotic error relation e_{n+1} \approx C e_n^\phi e_{n-1}^{1-\phi}, where e_n denotes the error at step n, leading to the characteristic equation p(p-1) = 1 whose positive solution is \phi. The asymptotic error constant C is given by C = \left| \frac{f''(\xi)}{2 f'(\xi)} \right|^{\phi - 1} for some \xi between the iterates, which is typically larger than the constant for Newton's method, reflecting the secant method's slower local convergence compared to the quadratic (order 2) behavior of Newton near the root. Convergence of the secant method requires initial approximations sufficiently close to the root such that the secant lines approach it monotonically, often ensured by bracketing the root (i.e., f(x_0) f(x_1) < 0) or selecting points where the function changes sign. The method is highly sensitive to the choice of starting points; if the initials are poorly selected, the iterates may diverge, particularly if successive secants fail to enclose the root or if the function is flat near the root. Despite its efficiency in avoiding derivative evaluations, the secant method converges more slowly than Newton's method in the final stages due to its sub-quadratic order, and it risks divergence or oscillation if the secant approximations overshoot the root repeatedly. To address these limitations, variants such as the modified secant method introduce a small fixed perturbation h to approximate the derivative via finite differences, preserving the order \phi while reducing sensitivity to exact secant slopes. Another acceleration, Steffensen's method, employs three function evaluations per step to simulate a divided-difference approximation equivalent to Newton's iteration, achieving quadratic convergence without explicit derivatives.

History and etymology

Origins of the term

The term "secant" derives from the Latin verb secare, meaning "to cut," reflecting its geometric connotation of a line that intersects or cuts through a curve, particularly a circle, at two points. This etymological root traces back to the Proto-Indo-European sek-, also the source of words like "saw" and "section," emphasizing division or severance. In mathematics, the term was first applied in the context of 16th-century trigonometry to denote a line segment from the center of a unit circle to a point on the circle, extended beyond to intersect the circle again, contrasting with the tangent's mere touch. The mathematical usage of "secant" was introduced by Danish mathematician Thomas Fincke in his 1583 treatise Geometriae rotundae, where he coined the term secans (Latin for "cutting") alongside tangens ("touching") to describe these trigonometric lines in spherical geometry and astronomy. Fincke's work marked the formal adoption of the term in European mathematical literature, building on earlier chord-based trigonometry but introducing a more intuitive nomenclature based on the line's interaction with the circle. This innovation facilitated computations in navigation and surveying, where such lines represented distances beyond the radius. In English, the term appeared in mathematical texts by the late 16th century, with Thomas Blundeville's 1594 M. Blundevile His Exercises providing the first published tables of sines, tangents, and secants, adapting Fincke's concepts for English readers in cosmography and navigation. By the 17th century, the concept extended from trigonometric circles to general geometric curves, as seen in early calculus where secant lines approximated tangents and derivatives, emphasizing their role in intersection and approximation. This broadening occurred amid the development of analytic geometry, where lines "cutting" arbitrary conics or algebraic curves became central to theorems like the power of a point. The term "secant" achieved modern standardization in English by the 18th century, appearing routinely in calculus textbooks such as those by Colin Maclaurin and in translations of continental works, solidifying its dual use in trigonometry and differential geometry.

Historical development

The concept of a secant, understood geometrically as a line intersecting a circle at two distinct points, originated implicitly in ancient Greek geometry. Euclid's Elements, composed around 300 BCE, explored such intersections extensively in Book III, including propositions on lines drawn from external points that cut the circle at two locations, forming the basis for later explicit developments without using the term itself. The trigonometric secant function emerged in the 16th century amid advances in tabular computation for astronomy and navigation. Regiomontanus (Johannes Müller) provided foundational tangent tables in his De triangulis omnimodis, published in 1533, which influenced subsequent work on reciprocal functions. Georg Joachim Rheticus introduced the secant as one of six trigonometric functions in his Canon doctrinae triangulorum (1551), marking its first printed appearance, while Francesco Maurolico independently computed and published a dedicated secant table (tabula benefica) in Theodosii sphaericorum elementorum libri tres (1558). Edmund Gunter further advanced practical use by incorporating logarithmic sines and tangents into his Canon triangulorum (1620, expanded 1624), facilitating computations in surveying and seafaring. During the 17th and 18th centuries, secant concepts integrated into the foundations of calculus. Isaac Newton employed finite differences, akin to secant slopes between points on a curve, as precursors to instantaneous rates of change in his fluxional calculus developed around 1665–1666. Gottfried Wilhelm Leibniz similarly used incremental differences in his differential calculus, formalized in publications from 1684, where secant-like approximations bridged finite changes to infinitesimals. The term "secant," derived from the Latin secāns (meaning "cutting"), aptly captured this geometric essence in early analytic contexts. In numerical analysis, the secant method for root-finding evolved from ancient false position techniques, with an iterative form described by Cardano in 1545. By the 19th century, secant values were standard in extensive trigonometric tables, such as those compiled for engineering and astronomy, enhancing computational accuracy.