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.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Secant Calculator
Calculator
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Formula: sec(x) = 1 / cos(x) | sec^2(x) = 1 + tan^2(x)
Worked example — sec(60) = 2.000000 | Identity verified: 4 = 4
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 sec(60) = 1 / cos(60) = 1 / 0.5 = 2 tan(60) = sqrt(3) = 1.7321 Verify: sec^2(60) = 4, 1 + tan^2(60) = 1 + 3 = 4 Identity 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 cos(theta) = 1/sqrt(2) = 0.7071 theta = arccos(0.7071) = 45 degrees Second solution: 360 - 45 = 315 degrees Both 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.
What is the inverse secant function (arcsecant)?
The inverse secant function, written as arcsec(x) or sec inverse of x, returns the angle whose secant equals the given value. Since secant values are always greater than or equal to 1 or less than or equal to -1, the domain of arcsecant is all real numbers with absolute value at least 1, which corresponds to the interval (-infinity, -1] union [1, infinity). The range is typically restricted to [0, pi] excluding pi/2 (or [0 degrees, 180 degrees] excluding 90 degrees) to make the function single-valued. Arcsecant can be computed using arccosine: arcsec(x) = arccos(1/x). This is useful because most calculators have an arccos button but not an arcsec button. The derivative of arcsec(x) is 1 / (|x| sqrt(x squared - 1)).
How does secant relate to the other five trigonometric functions?
Secant is interconnected with all other trigonometric functions through fundamental identities. As the reciprocal of cosine: sec(x) = 1/cos(x). Through the Pythagorean identity: sec squared(x) = 1 + tan squared(x). In terms of sine: sec(x) = 1 / sqrt(1 - sin squared(x)) for appropriate quadrants. In terms of tangent: sec(x) = plus or minus sqrt(1 + tan squared(x)). In terms of cosecant: sec(x) = csc(90 - x) due to the cofunction identity. In terms of cotangent: sec(x) = plus or minus sqrt(1 + 1/cot squared(x)). These relationships allow conversion between any pair of trigonometric functions, which is essential for simplifying complex trigonometric expressions, verifying identities, and solving equations where different functions appear together.
What are the special values of secant at common angles?
The secant values at standard angles can be derived from the well-known cosine values. At 0 degrees: sec(0) = 1/cos(0) = 1/1 = 1. At 30 degrees: sec(30) = 1/cos(30) = 2/sqrt(3), which is approximately 1.1547. At 45 degrees: sec(45) = 1/cos(45) = sqrt(2), approximately 1.4142. At 60 degrees: sec(60) = 1/cos(60) = 1/0.5 = 2. At 90 degrees: sec(90) is undefined because cos(90) = 0. At 120 degrees: sec(120) = -2. At 180 degrees: sec(180) = -1. These values repeat with period 360 degrees and are symmetric about 180 degrees. Memorizing these values is helpful for quickly evaluating trigonometric expressions without a calculator.
Why is secant less commonly taught than sine, cosine, and tangent?
Secant, along with cosecant and cotangent, is often considered a secondary trigonometric function because it can always be expressed as the reciprocal of cosine. Modern mathematics education tends to focus on the three primary functions (sine, cosine, tangent) since they suffice for most applications. However, secant remains important in several contexts. In calculus, sec squared(x) appears as the derivative of tan(x), making it essential for integration techniques. The secant function is also more convenient than writing 1/cos(x) repeatedly in complex expressions. Historically, before electronic calculators, secant was widely used in navigation and surveying because lookup tables for secant simplified certain calculations. In advanced mathematics, all six functions remain important for understanding the complete structure of trigonometric theory.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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