Secant Calculator
Free Secant Calculator for trigonometry. Enter values to get step-by-step solutions with formulas and graphs. Enter your values for instant results.
Calculator
Adjust values & calculateCommon Secant Values
Formula
The secant function is the reciprocal of cosine. It is undefined when cosine is zero (at 90 + 180k degrees). The Pythagorean identity sec^2(x) = 1 + tan^2(x) connects secant and tangent and is widely used in calculus.
Last reviewed: December 2025
Worked Examples
Example 1: Computing Secant of 60 Degrees
Example 2: Inverse Secant Calculation
Background & Theory
The Secant Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Secant Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
sec(x) = 1 / cos(x) | sec^2(x) = 1 + tan^2(x)
The secant function is the reciprocal of cosine. It is undefined when cosine is zero (at 90 + 180k degrees). The Pythagorean identity sec^2(x) = 1 + tan^2(x) connects secant and tangent and is widely used in calculus.
Worked Examples
Example 1: Computing Secant of 60 Degrees
Problem: Find sec(60 degrees) and verify using the Pythagorean identity.
Solution: cos(60) = 0.5\nsec(60) = 1 / cos(60) = 1 / 0.5 = 2\ntan(60) = sqrt(3) = 1.7321\nVerify: sec^2(60) = 4, 1 + tan^2(60) = 1 + 3 = 4\nIdentity sec^2 = 1 + tan^2 confirmed
Result: sec(60) = 2.000000 | Identity verified: 4 = 4
Example 2: Inverse Secant Calculation
Problem: Find the angle whose secant is sqrt(2).
Solution: sec(theta) = sqrt(2) = 1.4142\ncos(theta) = 1/sqrt(2) = 0.7071\ntheta = arccos(0.7071) = 45 degrees\nSecond solution: 360 - 45 = 315 degrees\nBoth give cos = 0.7071, sec = 1.4142
Result: arcsec(sqrt(2)) = 45 degrees (or 315 degrees)
Frequently Asked Questions
What is the secant function and how is it defined?
The secant function (sec) is the reciprocal of the cosine function, defined as sec(x) = 1 / cos(x). It is one of the six fundamental trigonometric functions, though it is less commonly used than sine, cosine, and tangent. Geometrically, in a right triangle, the secant of an angle equals the ratio of the hypotenuse to the adjacent side. On the unit circle, sec(x) represents the length of the line segment from the origin to the point where the terminal ray intersects the vertical line x = 1. The secant function is undefined wherever cosine equals zero, which occurs at 90 degrees plus any multiple of 180 degrees. Its range consists of all real numbers with absolute value greater than or equal to 1.
What is the Pythagorean identity involving secant?
The secant function participates in the Pythagorean identity: sec squared x = 1 + tan squared x. This is derived by dividing the fundamental identity sin squared x + cos squared x = 1 by cos squared x, yielding tan squared x + 1 = sec squared x. This identity is extensively used in calculus, particularly in trigonometric substitutions and integration. For example, the integral of sec squared x dx = tan x + C follows directly from recognizing sec squared x as the derivative of tan x. This identity also provides a way to compute secant from tangent and vice versa without needing to know the angle. It is one of three Pythagorean identities, along with sin squared + cos squared = 1 and 1 + cot squared = csc squared.
Where is the secant function undefined and why?
The secant function is undefined at angles where the cosine equals zero, because division by zero is undefined. These angles are 90 degrees, 270 degrees, and more generally, 90 + 180k degrees for any integer k (or pi/2 + k pi in radians). At these points, the secant function has vertical asymptotes, meaning the function values approach positive or negative infinity as the angle approaches these critical values. From the left of 90 degrees, secant approaches positive infinity, while from the right it approaches negative infinity. Understanding where secant is undefined is crucial for correctly graphing the function, solving equations involving secant, and establishing the domain of composite functions. The period of secant is 360 degrees (2 pi), the same as cosine.
How do you graph the secant function?
Graphing the secant function involves first understanding its relationship to cosine. Since sec(x) = 1/cos(x), the secant graph has vertical asymptotes wherever cosine equals zero. Start by sketching the cosine curve lightly. At every maximum of cosine (where cos = 1), secant also equals 1, creating a local minimum of the secant graph. At every minimum of cosine (where cos = -1), secant equals -1, creating a local maximum. Between these points and the asymptotes, the secant curve opens outward, forming U-shaped branches that extend toward infinity. The function has a period of 2 pi and exhibits even symmetry, meaning sec(-x) = sec(x). The resulting graph consists of alternating upward-opening and downward-opening parabola-like curves separated by vertical asymptotes.
What is the derivative and integral of the secant function?
The derivative of sec(x) is sec(x) tan(x). This can be derived using the quotient rule on 1/cos(x) or by applying the chain rule. The derivative is positive in quadrants I and IV (where both secant and tangent share the same sign) and negative in quadrants II and III. The integral of sec(x) dx is one of the more surprising results in calculus: it equals ln|sec(x) + tan(x)| + C, or equivalently ln|tan(x/2 + pi/4)| + C. This integral was historically important in cartography for computing the Mercator projection. The integral of sec squared(x) is simply tan(x) + C, which is more straightforward and appears frequently in problems involving integration and differential equations.
How is secant used in real-world applications?
The secant function appears in various practical applications across science and engineering. In optics, the secant of the angle of incidence determines how much a light beam spreads when passing through a medium at an oblique angle, affecting the path length through the medium. In surveying and navigation, secant corrections are applied to measured distances when terrain slopes are involved. The Mercator map projection uses the integral of secant to convert latitude lines to equally-spaced horizontal lines, making it essential for maritime navigation. In structural engineering, the secant formula describes the maximum deflection of a column under eccentric loading. In electrical engineering, secant appears in impedance calculations for AC circuits with reactive components.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy