Secant (Trig)
Secant is the reciprocal of cosine, defined as the ratio of the hypotenuse to the adjacent side in a right triangle.
Formula
\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}
Definition
Secant (written "sec") is the flip of cosine: $\sec(\theta) = 1/\cos(\theta) = \text{hypotenuse}/\text{adjacent}$. It is undefined when $\cos(\theta) = 0$ (at $90^\circ$, $270^\circ$), and because cosine is at most $1$ in size, secant is always at least $1$ in size too. The Pythagorean identity involving secant, $1 + \tan^2(\theta) = \sec^2(\theta)$, comes from dividing $\sin^2 + \cos^2 = 1$ through by $\cos^2$, and secant has the Taylor series $\sec(x) = 1 + x^2/2 + 5x^4/24 + 61x^6/720 + \ldots$ for $|x| < \pi/2$, whose coefficients are the Euler (secant) numbers.
Example
If $\cos(60^\circ) = 0.5$, then $\sec(60^\circ) = 1/0.5 = 2$; and $\sec(30^\circ) = 1/\cos(30^\circ) = 2/\sqrt{3} = 2\sqrt{3}/3 \approx 1.155$. The derivative $d/dx[\sec(x)] = \sec(x)\tan(x)$, and the integral of $\sec(x) = \ln|\sec(x) + \tan(x)| + C$, a non-obvious result used in the Mercator map projection formula.
Key Insight
The identity $1 + \tan^2(\theta) = \sec^2(\theta)$ appears frequently in calculus integration techniques. The Mercator projection maps latitude $\phi$ to $y = \ln|\sec(\phi) + \tan(\phi)|$, the integral of sec, which is why the integral of secant has historical significance: it was the key problem in 17th-century cartography.