Secant Calculator

What Is the Secant Function?

In trigonometry, the secant (abbreviated as $\sec$) is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function: $$\sec(\theta) = \frac{1}{\cos(\theta)}$$ In right-triangle trigonometry, for an acute angle $\theta$, the secant represents the ratio of the length of the hypotenuse to the length of the adjacent side: $$\sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{c}{b}$$ Because cosine can never exceed $1$ in magnitude ($|\cos(\theta)| \le 1$), the secant value of any real angle is always greater than or equal to $1$, or less than or equal to $-1$: $$|\sec(\theta)| \ge 1 \quad (\sec(\theta) \ge 1 \text{ or } \sec(\theta) \le -1)$$

Secant of Special Angles

The table below lists exact values of $\sec(\theta)$ for standard reference angles across the unit circle:

• $\theta = 0^\circ$ ($0$ rad): $\cos(0^\circ) = 1 \implies \sec(0^\circ) = 1$
• $\theta = 30^\circ$ ($\pi/6$ rad): $\cos(30^\circ) = \frac{\sqrt{3}}{2} \implies \sec(30^\circ) = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \approx 1.1547$
• $\theta = 45^\circ$ ($\pi/4$ rad): $\cos(45^\circ) = \frac{\sqrt{2}}{2} \implies \sec(45^\circ) = \sqrt{2} \approx 1.4142$
• $\theta = 60^\circ$ ($\pi/3$ rad): $\cos(60^\circ) = \frac{1}{2} \implies \sec(60^\circ) = 2$
• $\theta = 90^\circ$ ($\pi/2$ rad): $\cos(90^\circ) = 0 \implies \sec(90^\circ)$ is Undefined (Vertical Asymptote)
• $\theta = 180^\circ$ ($\pi$ rad): $\cos(180^\circ) = -1 \implies \sec(180^\circ) = -1$
• $\theta = 270^\circ$ ($3\pi/2$ rad): $\cos(270^\circ) = 0 \implies \sec(270^\circ)$ is Undefined

Fundamental Pythagorean Identity

The secant function is closely connected to the tangent function through the Pythagorean trigonometric identity: $$1 + \tan^2(\theta) = \sec^2(\theta) \implies \sec^2(\theta) - \tan^2(\theta) = 1$$ This identity allows mathematicians and engineers to convert between secant and tangent without directly evaluating the angle.

Domain, Range, and Asymptotes

Because $\sec(\theta) = \frac{1}{\cos(\theta)}$, the secant function is undefined wherever $\cos(\theta) = 0$. These zeros occur at odd multiples of $90^\circ$ (or $\frac{\pi}{2}$ radians): $$\theta \neq \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}$$ At these boundary lines, the graph of $y = \sec(x)$ exhibits vertical asymptotes approaching $+\infty$ or $-\infty$. You can check related angular reductions using our Reference Angle Calculator and find periodic repetitions with our Coterminal Angle Calculator.

Secant in Right Triangles

When working with right triangles, if you know the adjacent side $b$ and hypotenuse $c$, you can solve for secant directly: $$\sec(\theta) = \frac{c}{b}$$ $$\theta = \arccos\left(\frac{b}{c}\right)$$ For complete side and angle solutions of right triangles, explore our Right Triangle Calculator or solve compound angle relations using our Double Angle Formula Calculator.