# Secant Cosecant Calculator

Calculate secant, cosecant, and cotangent for any angle in degrees or radians. Shows all six trig values. Free online reciprocal trig calculator.

## What this calculates

Find sec, csc, and cot for any angle in degrees or radians. All six trig function values are displayed so you have the complete picture.

## Inputs

- **Angle** — The angle to evaluate.
- **Angle Unit** — options: Degrees, Radians — Choose degrees or radians.

## Outputs

- **Secant (sec θ)** — formatted as text — sec θ = 1 / cos θ
- **Cosecant (csc θ)** — formatted as text — csc θ = 1 / sin θ
- **Cotangent (cot θ)** — formatted as text — cot θ = cos θ / sin θ = 1 / tan θ
- **sin θ** — The sine of the angle (for reference).
- **cos θ** — The cosine of the angle (for reference).
- **tan θ** — formatted as text — The tangent of the angle (for reference).

## Details

Secant, cosecant, and cotangent are the reciprocal trigonometric functions. They are defined as the reciprocals of cosine, sine, and tangent respectively.

**Definitions:**

- **Secant:** sec θ = 1 / cos θ
- **Cosecant:** csc θ = 1 / sin θ
- **Cotangent:** cot θ = cos θ / sin θ = 1 / tan θ

**Common Values:**

| Angle | sec | csc | cot |
|-------|-----|-----|-----|
| 0° | 1 | Undefined | Undefined |
| 30° | 2/sqrt(3) = 1.1547 | 2 | sqrt(3) = 1.7321 |
| 45° | sqrt(2) = 1.4142 | sqrt(2) = 1.4142 | 1 |
| 60° | 2 | 2/sqrt(3) = 1.1547 | 1/sqrt(3) = 0.5774 |
| 90° | Undefined | 1 | 0 |

**When Are They Undefined?**

- sec θ is undefined when cos θ = 0 (at 90°, 270°, etc.)
- csc θ is undefined when sin θ = 0 (at 0°, 180°, 360°, etc.)
- cot θ is undefined when sin θ = 0 (same as csc)

**Identities Involving Reciprocal Functions:**

- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
- sec²θ - tan²θ = 1

These identities appear frequently in calculus, especially when integrating trigonometric expressions.

## Frequently Asked Questions

**Q: When would I use secant, cosecant, or cotangent instead of sin, cos, tan?**

A: Reciprocal functions come up naturally in calculus (the derivative of tan is sec², the integral of sec is ln|sec + tan|), physics (optics, wave equations), and engineering. They also simplify certain identities. For instance, writing 1 + tan²θ = sec²θ is cleaner than writing it in terms of sin and cos alone.

**Q: Why are secant and cosecant undefined at certain angles?**

A: Secant equals 1/cos θ, so when cos θ = 0 (at 90° and 270°), you are dividing by zero. Similarly, cosecant equals 1/sin θ, so it is undefined when sin θ = 0 (at 0°, 180°, 360°). At these points, the function values shoot off toward positive or negative infinity.

**Q: What is the Pythagorean identity for secant and tangent?**

A: The identity is 1 + tan²θ = sec²θ. It comes from dividing the fundamental identity sin²θ + cos²θ = 1 by cos²θ. This identity is heavily used in trigonometric substitution during integration.

**Q: How do I graph the secant and cosecant functions?**

A: Start by graphing cos θ (for sec) or sin θ (for csc). Wherever the base function crosses zero, draw a vertical asymptote. The reciprocal function curves away from those asymptotes. Secant has a U shape between each pair of asymptotes, while cosecant looks like inverted U shapes. Both functions never take values between -1 and 1.

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Source: https://vastcalc.com/calculators/math/secant-cosecant
Category: Math
Last updated: 2026-04-08