Tangent (Trig)
Tangent is a trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle.
Formula
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}
Definition
Tangent (written "tan") compares how tall a right triangle is to how wide it is: $\tan(\theta) = \text{opposite}/\text{adjacent} = \sin(\theta)/\cos(\theta)$. It is undefined when $\cos(\theta) = 0$ (at $90^\circ$ and $270^\circ$) and, unlike sine and cosine, can take any real value. It has period $\pi$ (not $2\pi$), vertical asymptotes at those excluded values, and the power series $\tan(x) = x + x^3/3 + 2x^5/15 + \ldots$ for $|x| < \pi/2$.
Example
If a tree casts a $10$-foot shadow and the sun makes a $40^\circ$ angle, $\tan(40^\circ) \approx 0.839$, so the tree height $= 10 \times 0.839 \approx 8.4$ feet. $\tan(45^\circ) = 1$ because the opposite and adjacent sides are equal there, while $\tan(89^\circ) \approx 57.3$ shows how steeply tangent grows near $90^\circ$. The derivative $d/dx[\tan(x)] = \sec^2(x)$ is always $\ge 1$, explaining why tan is strictly increasing on each branch, and its integral is $-\ln|\cos(x)| + C$.
Key Insight
Tangent tells you the "slope" of the angle, equal to the slope of the terminal side of the angle in standard position, $\text{rise}/\text{run} = \text{opposite}/\text{adjacent}$, directly connecting trig to coordinate geometry. In projective geometry, tangent naturally parametrizes the projective line, since it maps $(-\pi/2, \pi/2)$ bijectively onto $\mathbb{R}$, making it central to perspective transformations and Cayley-Klein metrics.