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Secant Calculator

Calculate the secant of any angle. Secant is the reciprocal of cosine: sec(θ) = 1/cos(θ).

Reviewed by Christopher FloiedPublished Updated

This free online secant calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

sec(θ)

1.154701

Angle (radians)

0.523599 rad

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Secant Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Secant Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Secant Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Secant Calculator

The secant function is the reciprocal of the cosine function: sec(θ) = 1/cos(θ). In a right triangle, it equals the ratio of the hypotenuse to the adjacent side. Secant is undefined at 90° and 270° (where cosine equals zero) and has a minimum absolute value of 1. The secant function has a period of 360° (2π radians) and is an even function: sec(−θ) = sec(θ). While secant is less commonly used in everyday calculations than sine, cosine, or tangent, it plays an important role in calculus, particularly in integration and trigonometric substitution. The integral of secant, ∫ sec(x) dx = ln|sec(x) + tan(x)| + C, is historically significant and was one of the key results needed for the Mercator map projection. This calculator computes the secant for any angle.

The Math Behind It

The secant function sec(θ) = 1/cos(θ) has vertical asymptotes at θ = 90° + n × 180° for any integer n. Its derivative is d/dx sec(x) = sec(x) tan(x), and the integral is ∫ sec(x) dx = ln|sec(x) + tan(x)| + C. The Pythagorean identity for secant is 1 + tan²(θ) = sec²(θ), which is fundamental in calculus. The integral ∫ sec²(x) dx = tan(x) + C is one of the most frequently used integration formulas. In trigonometric substitution, x = a sec(θ) is used for integrals involving √(x² − a²). The secant function appears in the Mercator projection formula: y = ln|sec(φ) + tan(φ)|, where φ is the latitude. This projection preserves angles (conformal) and maps rhumb lines (constant-bearing curves) as straight lines, making it invaluable for navigation. The secant numbers (Euler numbers Eₙ) count certain types of permutations and appear in the Taylor expansion of sec(x). Secant also appears in the relationship between the focal length and field of view of a camera lens.

Formula Reference

Secant Function

sec(θ) = 1 / cos(θ) = hypotenuse / adjacent

Variables: θ = angle

Worked Examples

Example 1: Secant at standard angles

Compute sec(60°).

Step 1:sec(60°) = 1/cos(60°)
Step 2:cos(60°) = 0.5
Step 3:sec(60°) = 1/0.5

sec(60°) = 2

Common Mistakes & Tips

  • !Trying to compute sec(90°) — it is undefined since cos(90°) = 0.
  • !Confusing secant with cosecant — sec = 1/cos, csc = 1/sin.
  • !Expecting sec(θ) to be between −1 and 1 — secant values are always |sec(θ)| ≥ 1 or undefined.

Related Concepts

Cosine Calculator

sec(θ) = 1/cos(θ).

Cotangent Calculator

Another reciprocal trig function.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Cotangent Calculator

Frequently Asked Questions

What is the range of the secant function?

sec(θ) can take any value in (−∞, −1] ∪ [1, ∞). It is never between −1 and 1.

Why is secant important in calculus?

The identity 1 + tan²(θ) = sec²(θ) appears in many integrals, and trigonometric substitution x = a sec(θ) handles integrals involving √(x² − a²).

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