Sec
Calculate secant of any angle in degrees. Free online secant calculator with step-by-step trigonometric results and interactive secant function graph.
About This Calculator
The secant function (sec) is one of the six fundamental trigonometric functions, defined as the reciprocal of cosine: sec(theta) = 1 / cos(theta). For any angle theta, it represents the ratio of the hypotenuse to the adjacent side in a right triangle. This free online secant calculator computes sec(theta) for any angle in degrees and displays an interactive graph of the secant function across a full 360-degree range.
To calculate secant, the calculator first converts the input angle from degrees to radians, computes the cosine using Math.cos, and then returns the reciprocal. The secant function has vertical asymptotes at angles where cos(theta) = 0 -- specifically at 90 deg and 270 deg (and their integer multiples). At these points, the secant is undefined (approaches infinity). Common values include sec(0 deg) = 1, sec(30 deg) ≈ 1.155, sec(45 deg) ≈ 1.414, and sec(60 deg) = 2.
Secant is widely used in calculus (the derivative of secant is sec(theta)tan(theta) and its integral is ln|sec(theta) + tan(theta)| + C), in physics (Snell's law for refraction), in electrical engineering (AC circuit analysis and impedance calculations), in geodesy (spherical trigonometry for great-circle navigation), and in optics (lens-maker's equation and paraxial ray tracing). Students studying trigonometry, precalculus, and calculus will find this tool useful for checking homework and visualizing the secant function.
Regional Notes
Trigonometry is taught worldwide as a core mathematics subject. In India, trigonometric functions including secant are introduced in Class 10 (CBSE/ICSE) and expanded in Classes 11-12. In the United States, secant is typically covered in Algebra 2, Precalculus, and Calculus courses. In the United Kingdom, it appears at A-Level Mathematics and Further Mathematics. Regardless of region, sec(theta) = 1/cos(theta) is the universal definition.
Frequently Asked Questions
What is secant in trigonometry?
Secant (sec) is the reciprocal of cosine. It is defined as sec(theta) = 1 / cos(theta). Secant is one of the six fundamental trigonometric functions and is used extensively in calculus, physics, and engineering.
How do you calculate secant?
To calculate secant of an angle: First find the cosine of the angle, then take its reciprocal. For example, sec(60 deg) = 1 / cos(60 deg) = 1 / 0.5 = 2. The secant function has asymptotes where cos(theta) = 0 (at 90 deg, 270 deg, etc.).
What is the range of secant?
The secant function has a range of (-∞, -1] union [1, ∞). It is never between -1 and 1. Secant approaches infinity as the angle approaches 90 deg (pi/2 rad) and 270 deg (3pi/2 rad), where cosine equals zero.
How does secant relate to other trig functions?
Secant is the reciprocal of cosine: sec(theta) = 1 / cos(theta). It relates to tangent through the Pythagorean identity: sec^2(theta) = 1 + tan^2(theta). Secant is also the reciprocal of the x-coordinate on the unit circle.
Where is secant used in real life?
Secant is used in physics (calculating refraction angles with Snell's law), in engineering (stress analysis and beam deflection), in optics (lens calculations), in navigation (spherical trigonometry for great-circle routes), and in calculus (derivative of secant is sec(theta)tan(theta)).
Is this secant calculator free?
Yes, all calculators on Calculy are completely free to use. No registration or download required.
What is the secant of 0 degrees?
The secant of 0 degrees is 1. This is because cos(0°) = 1, and sec(θ) = 1/cos(θ), so sec(0°) = 1/1 = 1.
Can I share my secant calculation results?
Yes, the calculator saves your input angle in the URL so you can bookmark or share the exact calculation with others. Simply copy the URL from your browser after calculating.