Welcome to Calcugo's secant calculator (or sec calculator, if you prefer), where we study the mysterious sec function. Arguably, it is one of the lesser-known functions in trigonometry, but it can still prove useful. Type an angle in degrees, radians, or gradians and the calculator returns sec(α) instantly — or switch modes and work backwards from a secant value to the angle (the arcsecant).
We will not only give the secant definition, i.e., the secant formula, but we'll also discuss what the domain of secant is, give the sec graph, and explain how to find the secant using alternative methods. So what is this secant equal to? Let's move to the first section and find out!
Sec in Math: The Secant Definition
All of trigonometry begins with a right triangle (you know, one of those that the Pythagorean theorem is about). Trigonometric functions, such as the one in our sec calculator, are a tool to describe the correspondences between a triangle's sides. In particular, we have there the secant definition.
So what is secant equal to? The secant of an angle is the hypotenuse divided by the side (leg) next to the angle (the adjacent leg). In other words, it is the reciprocal of the cosine:
sec(α) = hypotenuse / adjacent = c / b = 1 / cos(α)
A keen eye will observe that we have no information about the size of the triangle. That is precisely the magic behind trigonometric functions such as sec in math: even if we scale the triangle to a different size, as long as the angle stays the same, the value of the function will not change.
Despite this advantage, there is a slight drawback: with the sec trig function defined this way (and all the other functions, for that matter), we can only have an angle between 0 and 90 degrees (or between 0 and π/2 in radians), simply because it's a right triangle. There is, however, a way to extend the whole thing to any real angle, even a negative one. The trick is to translate the secant definition into a two-dimensional Euclidean plane.
Let A = (x, y) be a point on the Euclidean plane, and denote by α the angle going counterclockwise from the positive half of the horizontal axis to the line segment connecting (0, 0) and A. (We say α goes from one line to the other, not that it is between them — because of that, we often call α a directed angle.) Such an angle can be larger than 90 degrees, larger than a full 360 degrees, and — since it is directed — even negative when we go clockwise instead of counterclockwise.
For such angles, we extend the definition by substituting a → y, b → x, and c → √(x² + y²) (the distance from the origin to A). Comparing with the secant formula above, we obtain the general secant formula:
sec(α) = √(x² + y²) / x
This extends the secant to any real angle — even negative ones and angles beyond a full turn of 360°. Since cosine appears in the denominator, sec(α) is undefined whenever x = 0, i.e., at α = 90°, 270°, 450°, … (every odd multiple of 90°), where the cosine equals zero.
Sec Function Graph, the Domain of Secant
You might have already seen the graph of sine or that of tangent, since they are among the most commonly used trigonometric functions. The sec graph, however, is rarely shown at school, and it's a real pity, because it's quite funky, to say the least.
Because sec(α) = 1/cos(α), the secant graph is the reciprocal of the cosine wave. Wherever cos(α) = 0, the secant shoots off to ±∞, producing vertical asymptotes at 90°, 270°, and so on (every odd multiple of 90°). Between the asymptotes, the curve forms a series of U-shaped branches. The secant never takes a value strictly between −1 and 1: its range is sec(α) ≤ −1 or sec(α) ≥ 1, and each branch has a minimum of 1 (at 0°, ±360°, …) or a maximum of −1 (at 180°, 180°±360°, …). The domain of secant is therefore every real number except the odd multiples of 90°. Like cosine, the secant is periodic with period 2π and is an even function: sec(−α) = sec(α).
How to Find the Secant — Alternative Secant Formulas
Depending on what you already know, there is more than one way to reach the secant:
- From cosine (reciprocal): sec(α) = 1 / cos(α). This is the definition used by the calculator.
- From a right triangle (SOHCAHTOA): sec(α) = hypotenuse / adjacent.
- From coordinates: sec(α) = √(x² + y²) / x for a point (x, y) on the terminal side of the angle.
- From the Pythagorean identity: sec²(α) = 1 + tan²(α), so sec(α) = ±√(1 + tan²(α)) (the sign follows the quadrant).
- Cofunction identity: sec(α) = csc(90° − α) — the secant of an angle equals the cosecant of its complement.
Secant Values Table (sec 0, sec 30°, sec 45°…)
The exact secant value is easy to remember for a handful of standard angles. Note the “undef” entries at 90° and 270°, where the cosine is zero.
| α (degrees) | α (radians) | sec(α) — exact | sec(α) — decimal |
|---|---|---|---|
| 0° | 0 | 1 | 1 |
| 30° | π/6 | 2√3/3 | 1.1547005384 |
| 45° | π/4 | √2 | 1.4142135624 |
| 60° | π/3 | 2 | 2 |
| 90° | π/2 | undefined | ±∞ |
| 120° | 2π/3 | −2 | −2 |
| 135° | 3π/4 | −√2 | −1.4142135624 |
| 150° | 5π/6 | −2√3/3 | −1.1547005384 |
| 180° | π | −1 | −1 |
| 270° | 3π/2 | undefined | ±∞ |
| 360° | 2π | 1 | 1 |
Remember the periodicity of the secant — sec(α + 360°) = sec(α) — if your angle is not shown in the table above.
Example: Using the Secant Calculator
- Keep the mode on Angle → secant.
- Enter the angle in the input field. For example, type 60.
- Choose the angle unit — degrees, radians, or gradians.
- Click Calculate. The calculator instantly shows sec(60°) = 2.
- It also shows the same angle converted to every unit, the related trig values (cos, sin, csc, tan, cot), and the quadrant the angle lands in.
Want to go the other way? Switch the mode to Secant value → angle, type a number with absolute value at least 1 (for example 2), and the calculator returns the angle in the [0°, 180°] range, excluding 90° (this is the arcsecant, or inverse secant).
FAQs
What is the secant of 0, 30, 45, and 60 degrees?
sec 0° = 1, sec 30° = 2√3/3 ≈ 1.1547, sec 45° = √2 ≈ 1.4142, and sec 60° = 2. The secant of 90° and 270° is undefined because the cosine of those angles is zero.
Is secant an even or odd function?
Secant is an even function, meaning sec(−α) = sec(α). Its graph is symmetric about the vertical axis, just like the cosine it is built from.
What is the domain and range of the secant function?
The domain of secant is every real number except odd multiples of 90° (90°, 270°, …), where the cosine is zero. Its range is sec(α) ≤ −1 or sec(α) ≥ 1 — the secant never lies strictly between −1 and 1. That is why the inverse mode only accepts values with |x| ≥ 1.
How are secant and cosine related?
They are reciprocals: sec(α) = 1/cos(α), and equivalently cos(α) = 1/sec(α). Wherever cosine is zero, the secant is undefined, and where cosine reaches ±1 the secant also reaches ±1.
How do I convert between degrees, radians, and gradians?
A full circle is 360° = 2π rad = 400 grad. To convert degrees to radians, multiply by π/180; to convert degrees to gradians, multiply by 10/9. This calculator shows all three units automatically.