# Secant Function — Formula, Graph, Properties, Examples

TL;DR

The secant function \( \sec \theta = \frac{1}{\cos \theta} \) is the reciprocal of cosine — defined wherever cosine is non-zero, with vertical asymptotes at \( \theta = (2n+1)\frac{\pi}{2} \) and range \((-, -1] \cup [1, )\). This article gives the formula, the graph paired with cosine, the table of values at special angles, the even-function symmetry, three worked examples in degrees and radians, and the common mistakes.

The **secant function** is defined as the multiplicative reciprocal of the cosine function. For any angle \( \theta \) where \( \cos \theta \neq 0 \):  
\[ \sec \theta = \frac{1}{\cos \theta}. \]

The function inherits its sign pattern, its period, and its domain restrictions from cosine — but its range and graphical shape are very different.

## The Formula and Triangle Definition

\[ \sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}} \]

In a right triangle with one acute angle \( \theta \), the secant is the ratio of the hypotenuse to the adjacent leg — the reciprocal of the cosine ratio (adjacent over hypotenuse).

On the unit circle, where a point at angle \( \theta \) has coordinates \((\cos \theta, \sin \theta)\):  
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\text{x-coordinate}}. \]

> **Quick facts.**
> 
> - **Reciprocal partner:** cosine, \( \sec \theta \cdot \cos \theta = 1 \).  
> - **Domain:** \( \theta \in \mathbb{R}, \theta \neq (2n+1)\frac{\pi}{2} \). In degrees, \( \theta \neq 90^{\circ}, 270^{\circ}, 450^{\circ}, \ldots \)  
> - **Range:** \((-, -1] \cup [1, )\). The value \( |\sec \theta| \ge 1 \) everywhere it's defined.  
> - **Period:** \( 2\pi \).  
> - **Symmetry:** even function — \( \sec(-\theta) = \sec \theta \).  
> - **Special values:** \[ \sec 0 = 1, \quad \sec\left(\frac{\pi}{3}\right) = 2, \quad \sec\left(\frac{\pi}{4}\right) = \sqrt{2}, \quad \sec\left(\frac{\pi}{6}\right) = \frac{2}{\sqrt{3}}, \quad \sec\left(\frac{\pi}{2}\right) \text{ undefined} \].  
> - **Derivative:** \[ \frac{d}{d\theta} \sec \theta = \sec \theta \tan \theta \].

## Double-Anchoring — Right Triangle and Unit Circle

The same secant value reads cleanly two ways.

**From the right triangle.** Consider a 30−60−90 triangle with sides 1, \( \sqrt{3} \), 2 — the side opposite 30° is 1, the side opposite 60° is \( \sqrt{3} \), and the hypotenuse is 2. For \( \theta = 60^{\circ} \):

- \( \cos 60^{\circ} = \frac{1}{2} \) 
- \( \sec 60^{\circ} = 2. \)

**From the unit circle.** At \( \theta = \frac{\pi}{3} \) (i.e., 60°), the unit-circle point is \((\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})) = (\frac{1}{2}, \frac{\sqrt{3}}{2})\). So:

\[ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}} = 2. \]

## The Graph of \( \sec \theta \)

The secant graph is most easily understood by overlaying it on the cosine graph.

- Where \( \cos \theta = 1 \) (at \( \theta = 0, 2\pi,\ldots \)), \( \sec \theta = 1 \) — the secant graph touches the cosine graph from above.  
- Where \( \cos \theta = -1 \) (at \( \theta = \pi, 3\pi, \ldots \)), \( \sec \theta = -1 \) — the secant graph touches from below.  
- Where \( \cos \theta = 0 \) (at \( \theta = \frac{\pi}{2},\frac{3\pi}{2},\ldots \)), \( \sec \theta \) is undefined — vertical asymptote.  
- Between the asymptotes, the secant graph is a "U" opening upward (in regions where cosine is positive) or a "U" opening downward (where cosine is negative).

### Table of Special Values

| \( \theta \) (rad) | \( \theta \) (deg) | \( \cos \theta \) | \( \sec \theta \) |
| --- | --- | --- | --- |
| 0 | 0° | 1 | 1 |
| \( \frac{\pi}{6} \) | 30° | \( \frac{\sqrt{3}}{2} \) | \( \frac{2}{\sqrt{3}} \) |
| \( \frac{\pi}{4} \) | 45° | \( \frac{\sqrt{2}}{2} \) | \( \sqrt{2} \) |
| \( \frac{\pi}{3} \) | 60° | \( \frac{1}{2} \) | 2 |
| \( \frac{\pi}{2} \) | 90° | 0 | undefined |
| \( \frac{2\pi}{3} \) | 120° | -\( \frac{1}{2} \) | -2 |
| \( \frac{3\pi}{4} \) | 135° | -\( \frac{\sqrt{2}}{2} \) | -\( \sqrt{2} \) |
| \( \frac{5\pi}{6} \) | 150° | -\( \frac{\sqrt{3}}{2} \) | -\( \frac{2}{\sqrt{3}} \) |
| \( \pi \) | 180° | -1 | -1 |
| \( \frac{3\pi}{2} \) | 270° | 0 | undefined |

## Properties of Secant Function

- **Even function.** \( \sec(-\theta) = \sec \theta \). The graph is symmetric about the \( y \)-axis.  
- **Periodic with period \( 2\pi \)**. \( \sec(\theta + 2\pi) = \sec \theta \).  
- **Range outside \((-1,1)\)**. The function never takes a value strictly inside \((-1,1)\).  
- **Has no amplitude.** Unlike sine and cosine, secant is unbounded — no "max" or "min" value.  
- **Pythagorean form:** \( \sec^2\theta - \tan^2\theta = 1 \).  
- **Reciprocal Pythagorean:** \( \sec^2\theta = 1 + \tan^2\theta \).

## Three Worked Examples of Secant Function

**Quick.** Compute \( \sec(\frac{\pi}{3}) \).

By the reciprocal identity:
\[ \sec(\frac{\pi}{3}) = \frac{1}{\cos(\frac{\pi}{3})} = \frac{1}{\frac{1}{2}} = 2. \]

In degrees, \( \frac{\pi}{3} = 60^{\circ} \), so \( \sec 60^{\circ} = 2 \).

**Final answer:** \( \sec(\frac{\pi}{3}) = \sec 60^{\circ} = 2 \).

**Standard.** Given \( \tan \theta = \frac{5}{12} \), find \( \sec \theta \).

_The wrong path._ A student writes "sec and tan are reciprocals" — they aren't — and concludes \( \sec \theta = \frac{12}{5} \).

The flaw: **the three reciprocal pairs are (sin, csc), (cos, sec), (tan, cot)**. Sec pairs with cos, not with tan.

The rescue. Apply \( \sec^2\theta = 1 + \tan^2\theta \):  
\[ \sec^2\theta = 1 + \left(\frac{5}{12}\right)^2 = 1 + \frac{25}{144} = \frac{169}{144}. \]

Since \( \theta \) is in the first quadrant, \( \sec \theta > 0 \), so \( \sec \theta = \frac{13}{12} \).

**Final answer:** \( \sec \theta = \frac{13}{12} \).

**Stretch.** Find the area of one full "smile" of the secant graph between two consecutive vertical asymptotes.

The antiderivative of \( \sec \theta \) is \( \ln|\sec \theta + \tan \theta| + C \).

As \( \epsilon \to 0^{+} \), both \( \sec(\frac{\pi}{2}-\epsilon) \) and \( \tan(\frac{\pi}{2}-\epsilon) \to +\infty \), so the integral \( \int_{0}^{\frac{\pi}{2}} \sec \theta \, d\theta = +\infty \).

## Where Secant Earns Its Place in the Toolkit

The secant function shows up in any setting where a ratio of total-to-projected length matters.

- **Mercator projection in cartography.** The Mercator world map scales latitude by \( \sec \phi \) where \( \phi \) is the latitude.  
- **Optics — refraction angles at glancing incidence.** Snell's law behaves like \( \sec \theta \).  
- **Tilted-axis sensor calibration.** A solar panel tilted receives flux proportional to \( \sec \theta \).  
- **Calculus — the antiderivative \( \int \sec \theta \, d\theta \)**.  
- **Beam structures and shear.** In a tilted beam, the secant of the tilt angle computes the effective load-bearing thickness.

## A Brief History of Secant Function

Edmund Gunter coined _secant_ in his _Canon Triangulorum_ (1620) — from the Latin _secare_, to cut. He referred to it as the length of the line segment cutting from a circle's centre through the angle's endpoint to a tangent line.

## Where Things Go Sideways With Secant Function

**Common mistakes:**
1. Confusing \( \sec \theta \) with \( \sin \theta \).
2. Forgetting the asymptotes when sketching the graph.
3. Including \( \sec \theta \) values in \((-1, 1)\).
4. Mixing degree-mode and radian-mode evaluation.

## Wrapping Up

- The **secant function** \( \sec \theta = \frac{1}{\cos \theta} \) is the reciprocal of cosine — undefined wherever cosine is zero.
- The domain is \( \theta \neq (2n+1)\frac{\pi}{2} \); the range is \((-, -1] \cup [1, )\); the period is \( 2\pi \).
- The graph is a sequence of U-shaped branches separated by vertical asymptotes, with \( |y| \ge 1 \) everywhere.