Secant (sec)

Secant $\sec\theta = \frac{1}{\cos\theta}$.

Domain: $\theta \neq \pi/2 + k\pi$. Range: $|\sec\theta| \geq 1$.

Right triangle: $\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}}$.

Pythagorean identity: $1 + \tan^2\theta = \sec^2\theta$ — useful in calculus integrals (e.g. trig substitution involving $\sqrt{a^2 + x^2}$).

Derivative: $\frac{d}{dx}\sec x = \sec x \tan x$.

Integral: $\int \sec x \, dx = \ln|\sec x + \tan x| + C$ — surprisingly tricky; the standard textbook trick is to multiply by $\frac{\sec x + \tan x}{\sec x + \tan x}$.

Secant has vertical asymptotes at every multiple of $\pi/2$ where cosine is zero, with U-shapes between asymptotes. Modern usage mostly via the integral / derivative formulas; for arithmetic, students convert to $1/\cos$.