Graph of the Tangent Function $y = \tan x$

The tangent function $y = \tan x$ is the sine divided by the cosine¹. On the unit circle it is the slope of the line through the origin at angle $x$, and in a right triangle it is the ratio of the opposite side to the adjacent side.

Definition

The denominator is $\cos x$, so the function is undefined wherever the cosine vanishes.

Domain and range

• The domain is every real number with $x \neq \dfrac{\pi}{2} + n\pi$
• The range is all real numbers
• The period is $\pi$
• It is an odd function

Unlike the sine and cosine, the tangent has no upper or lower bound.

Periodicity

The period is $\pi$, half that of the sine and cosine, since $\tan(x + \pi) = \tan x$. The same branch repeats every $\pi$.

Symmetry

Since $\tan(-x) = -\tan x$, it is an odd function with point symmetry about the origin.

Asymptotes and limits

There is a vertical asymptote at each $x = \dfrac{\pi}{2} + n\pi$. Which way the value flies off depends on the sign the denominator approaches from.

The graph is thus a family of branches, each penned in between two asymptotes.

Monotonicity

The derivative is $y' = \dfrac{1}{\cos^2 x} = \sec^2 x$, positive wherever the function is defined, so the tangent increases monotonically within each branch.

It is not increasing on the real line as a whole: crossing an asymptote, the value drops from $+\infty$ back to $-\infty$. The increase holds only inside a single branch.

Notable values

The zeros are the solutions of $\sin x = 0$, namely $x = n\pi$. At $\dfrac{\pi}{4}$, that is $45^\circ$, the value is exactly $1$, which corresponds to a line of slope $1$.

Approximation near the origin

The tangent line at the origin is $y = x$, of slope $1$.

So $\tan x \approx x$ for small angles. Writing one more term of the expansion gives $\tan x = x + \dfrac{x^3}{3} + \cdots$, so near the origin $\sin x$ falls below $x$ while $\tan x$ rises above it.

Relationships with other functions

It satisfies $1 + \tan^2 x = \dfrac{1}{\cos^2 x}$, and its reciprocal is the cotangent $\cot x$. Because its range is the whole real line, restricting it to $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$ makes it one-to-one, and the inverse of that restriction is $\arctan x$.

Applications

The angle $\theta$ that the line $y = mx$ makes with the $x$-axis satisfies $m = \tan\theta$, so the tangent is the bridge between slope and angle.

• The gradient of a road or a roof
• Surveying, where a height is a distance times $\tan\theta$
• The relation between the field of view of a camera and its focal length

1. Trigonometric functions, Wikipedia