Tangent (Trig)
TrigonometryTangent 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. It is opposite divided by adjacent.
Example
If a tree casts a shadow of $10$ feet 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.
Key Insight
Tangent tells you the "slope" of the angle. A steep angle has a big tangent; a shallow angle has a small tangent.
Definition
$\tan(\theta) = \text{opposite}/\text{adjacent} = \sin(\theta)/\cos(\theta)$. Tangent is undefined when $\cos(\theta) = 0$ (at $90^\circ$ and $270^\circ$). It can take any real value, unlike sine and cosine which are bounded between $-1$ and $1$.
Example
$\tan(45^\circ) = 1$ because at $45^\circ$ the opposite and adjacent sides are equal. $\tan(89^\circ) \approx 57.3$, showing how steeply tangent grows near $90^\circ$.
Key Insight
Tangent equals the slope of the terminal side of the angle in standard position: $\text{slope} = \text{rise}/\text{run} = \text{opposite}/\text{adjacent}$. This directly connects trig to coordinate geometry.
Definition
$\tan(x) = \sin(x)/\cos(x)$, defined for $x$ not equal to $\pi/2 + n\pi$. 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
The derivative $d/dx[\tan(x)] = \sec^2(x)$, which is always $\ge 1$, explaining why tan is a strictly increasing function on each branch. The integral of $\tan(x) = -\ln|\cos(x)| + C$.
Key Insight
In projective geometry, the tangent function naturally parametrizes the projective line, since tan maps $(-\pi/2, \pi/2)$ bijectively onto $\mathbb{R}$. This makes it central to perspective transformations and Cayley-Klein metrics.