Secant Calculator

What this calculator does

So, you have an angle and you want its secant. This tool takes the angle, in degrees or radians, and returns the secant rounded to three decimal places, or tells you when it is not defined.

One box for the angle, a dropdown for degrees or radians, and a Calculate button.

How to use it

1. Type your angle.
2. Choose degrees or radians to match it.
3. Press Calculate.

Match the unit to your value. At the angles where secant has no value, the tool returns "Not defined" rather than a runaway number.

Secant is one over cosine

Secant is not a new measurement to learn, it is simply the reciprocal of cosine. That is exactly how the tool computes it: it works out the cosine of your angle and then divides 1 by it.

sec(angle) = 1 / cos(angle)

So everywhere cosine is large, secant is small, and everywhere cosine is small, secant is large. If you can find a cosine, you already know how to find a secant: flip it. Secant belongs to the group of reciprocal trigonometric functions, alongside cosecant, which flips sine, and cotangent, which flips tangent.

Why secant is never between -1 and 1

Cosine always lands between -1 and 1. Flipping a number in that range does something specific: dividing 1 by a value no bigger than 1 always gives a result at least as big as 1. So secant does the opposite of cosine, it never sits inside the band between -1 and 1, and instead always has size 1 or more. When cosine is at its largest, 1, secant is also 1. As cosine shrinks toward zero, secant grows without any upper limit.

Where secant is undefined

Because secant divides by cosine, it runs into trouble wherever cosine is zero. That happens at 90 degrees, at 270 degrees, and at every odd multiple of 90. At those angles you would be dividing 1 by 0, which has no meaning, so the secant is undefined there. The tool watches for exactly these angles and reports "Not defined", the same points where the tangent also breaks down, since both depend on cosine sitting underneath.

A worked example

Take 60 degrees. The cosine of 60 degrees is 0.5, so the secant is 1 divided by 0.5, which is 2, shown as 2.000. At 0 degrees the cosine is 1, so the secant is also 1. And at 90 degrees, where the cosine is 0, the tool returns Not defined.

Where secant shows up

Secant is less of an everyday face than sine or cosine, but it earns its place, especially in calculus. It appears whenever a problem naturally produces one over a cosine rather than a cosine, and it has a tidy identity worth knowing: 1 plus tangent squared equals secant squared, which ties it directly to the tangent. So while you might reach for sine and cosine first, secant is the clean way to write and work with expressions that would otherwise be cluttered with cosines in the denominator.

Questions people ask

What is the secant of an angle?

The reciprocal of the cosine: 1 divided by the cosine of that angle. Large where cosine is small, and small where cosine is large.

Why is secant never between -1 and 1?

Because cosine always is, and dividing 1 by a number no larger than 1 gives a result of size at least 1. So secant sits at 1 or beyond, never inside that band.

When is secant undefined?

Wherever cosine is zero, which is 90 degrees, 270 degrees, and every odd multiple of 90. There you would divide by zero, so no value exists.

Do I need cosine to find secant?

In effect, yes. Secant is one over cosine, so the cosine value is exactly what gets flipped to give the secant.

What are the other reciprocal functions?

Cosecant is one over sine, and cotangent is one over tangent. Secant is the one over cosine of that set.

References

On secant as the reciprocal of cosine. Secant is defined as 1 over the cosine, one of the reciprocal circular functions, undefined where cosine vanishes.

1. Eric W. Weisstein, "Trigonometric Functions," from MathWorld, a Wolfram resource, on the secant and the full set of circular functions.
2. Eric W. Weisstein, "Trigonometry," from MathWorld, a Wolfram resource, on the unit-circle definitions relating secant to cosine.