Secant (Trig)
TrigonometrySecant 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. Instead of adjacent over hypotenuse, it is hypotenuse over adjacent.
Example
If $\cos(60^\circ) = 0.5$, then $\sec(60^\circ) = 1/0.5 = 2$.
Key Insight
Just like cosecant is always at least $1$ in size, secant is always at least $1$ in size too, because you are flipping a fraction that was already $1$ or less.
Definition
$\sec(\theta) = 1/\cos(\theta) = \text{hypotenuse}/\text{adjacent}$. It is undefined when $\cos(\theta) = 0$ (at $90^\circ$, $270^\circ$). The Pythagorean identity involving secant is: $1 + \tan^2(\theta) = \sec^2(\theta)$.
Example
$\sec(30^\circ) = 1 / \cos(30^\circ) = 1 / (\sqrt{3}/2) = 2/\sqrt{3} = 2\sqrt{3}/3 \approx 1.155$.
Key Insight
The identity $1 + \tan^2(\theta) = \sec^2(\theta)$ is derived by dividing $\sin^2 + \cos^2 = 1$ through by $\cos^2$. This identity appears frequently in calculus integration techniques.
Definition
$\sec(x) = 1/\cos(x)$ has period $2\pi$, poles where $\cos(x) = 0$, and the Taylor series $\sec(x) = 1 + x^2/2 + 5x^4/24 + 61x^6/720 + \ldots$ for $|x| < \pi/2$. The coefficients are the Euler (secant) numbers.
Example
The derivative $d/dx[\sec(x)] = \sec(x)\tan(x)$. 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 Mercator projection maps latitude $\phi$ to $y = \ln|\sec(\phi) + \tan(\phi)|$, the integral of sec. This is why the integral of secant has historical significance: it was the key problem in 17th-century cartography.