Secant Calculator - sec(x)
Enter an angle and select your unit (degrees, radians, or gradians) to find its secant instantly. You also get the inverse: type a secant value to find the arcsecant angle. The step-by-step panel shows the full working, and the curve visual plots where your angle falls on the sec(x) graph.
Formula
Worked example
Find sec(60 deg): cos(60 deg) = 0.5, so sec(60 deg) = 1/0.5 = 2. Exact form: 2. For the inverse, arcsec(2) = arccos(1/2) = 60 deg.
What is the secant function?
The secant function, written sec(x), is one of the six fundamental trigonometric functions. It is the reciprocal of the cosine: sec(x) = 1 / cos(x). In a right-angled triangle, sec(x) equals the hypotenuse divided by the adjacent side, which is the same ratio that cosine flips upside-down. Because cosine never exceeds 1 in absolute value, its reciprocal the secant is always outside the range (-1, 1), meaning sec(x) is always >= 1 or <= -1. This is why the secant curve has a distinctive shape with two separate branches that never cross the horizontal band between -1 and 1.
How to calculate sec(x)
The fastest route is always sec(x) = 1 / cos(x). Your calculator or software has a cosine function built in, so compute cos(x), then take its reciprocal. Make sure you are in the correct angle mode - degrees, radians, or gradians - before you start. At the special angles you can use exact forms: sec(0 deg) = 1, sec(30 deg) = 2√3/3, sec(45 deg) = √2, sec(60 deg) = 2, sec(90 deg) is undefined, sec(180 deg) = -1. These exact values come from the exact cosines of those angles and are useful in proofs and examinations.
Inverse secant (arcsecant)
The inverse secant function, arcsec(x), answers the question: which angle has a secant equal to x? Because secant is periodic (repeating every 360 deg or 2pi rad), there are infinitely many such angles. By convention, arcsec(x) returns the principal value: a result in [0 deg, 90 deg) when x >= 1, and in (90 deg, 180 deg] when x <= -1. The calculator also shows the alternate angle in the full 0-360 deg cycle, which is 360 deg minus the primary result. The computation is arcsec(x) = arccos(1/x), which converts it to the standard arccos that any computer or calculator provides.
Secant identities and the sec(x) graph
A handful of identities link secant to the other trig functions. The Pythagorean identity in secant form is sec^2(x) = 1 + tan^2(x). Secant is also equal to tan(x)/sin(x), and it relates to the hypotenuse-to-adjacent ratio in a right triangle. The graph of y = sec(x) consists of U-shaped branches that open upward (for cos > 0) and downward (for cos < 0), with vertical asymptotes at every angle where cos(x) = 0, namely x = 90 deg, 270 deg, and their equivalents in each full cycle. The function has a period of 360 deg (2pi radians) and is an even function, meaning sec(-x) = sec(x).
Secant values for common angles
| Angle (deg) | Angle (rad) | sec(x) exact | sec(x) decimal |
|---|---|---|---|
| 0 | 0 | 1 | 1.000000 |
| 15 | π/12 | √6 - √2 | 1.035276 |
| 30 | π/6 | 2√3/3 | 1.154701 |
| 45 | π/4 | √2 | 1.414214 |
| 60 | π/3 | 2 | 2.000000 |
| 75 | 5π/12 | √6 + √2 | 3.863703 |
| 90 | π/2 | undefined | undefined |
| 120 | 2π/3 | -2 | -2.000000 |
| 135 | 3π/4 | -√2 | -1.414214 |
| 150 | 5π/6 | -2√3/3 | -1.154701 |
| 180 | π | -1 | -1.000000 |
| 210 | 7π/6 | -2√3/3 | -1.154701 |
| 225 | 5π/4 | -√2 | -1.414214 |
| 240 | 4π/3 | -2 | -2.000000 |
| 270 | 3π/2 | undefined | undefined |
| 300 | 5π/3 | 2 | 2.000000 |
| 315 | 7π/4 | √2 | 1.414214 |
| 330 | 11π/6 | 2√3/3 | 1.154701 |
| 360 | 2π | 1 | 1.000000 |
Exact and decimal secant values at the standard special angles. Undefined at 90 deg, 270 deg, etc.
Frequently asked questions
What is secant in trigonometry?
Secant (sec) is the reciprocal of cosine. If cos(x) gives the ratio of adjacent to hypotenuse in a right triangle, sec(x) = 1/cos(x) gives hypotenuse over adjacent. Its value is always >= 1 or <= -1, since cosine is bounded between -1 and 1.
How do I calculate sec(x) on a calculator that does not have a sec button?
Enter your angle, press the cos button, then take the reciprocal (usually the 1/x or x^-1 key). For example, sec(60 deg): cos(60 deg) = 0.5, then 1/0.5 = 2. Always confirm your calculator is in the correct angle mode (degrees or radians) first.
Why is sec(90 deg) undefined?
cos(90 deg) = 0, and dividing 1 by 0 is undefined. Geometrically, at 90 deg the adjacent side in a right triangle has zero length, so the hypotenuse-to-adjacent ratio has no finite value. On the graph, this appears as a vertical asymptote.
What values can sec(x) not take?
Secant is never strictly between -1 and 1. It can equal exactly 1 (at 0 deg, 360 deg, ...) or exactly -1 (at 180 deg, ...), but any value in the open interval (-1, 1) is impossible. This is because |cos(x)| <= 1 means |1/cos(x)| >= 1.
What is arcsec(x) and how is it different from 1/sec(x)?
arcsec(x) is the inverse secant function: it takes a secant value and returns the angle. For example, arcsec(2) = 60 deg because sec(60 deg) = 2. It is not the same as 1/sec(x), which just equals cos(x). The superscript -1 notation (sec^-1) means inverse function here, not reciprocal.
How does secant differ from cosecant?
Secant is 1/cos(x) (hypotenuse over adjacent), while cosecant (csc) is 1/sin(x) (hypotenuse over opposite). They are reciprocals of different primary trig functions and produce different graphs and values except at special angles where they coincide by symmetry.