The Tangent Function

Definition For every real $x$ with $\cos x \ne 0$, the tangent of $x$ is

Since $P(x) = (\cos x, \sin x)$ is the point on the unit circle at angle $x$ radians, $\tan x$ is the slope of the line through the origin and $P(x)$. Its sign follows from the quadrant signs, positive in quadrants I and III and negative in quadrants II and IV.

Definition The quotient $\frac{\sin x}{\cos x}$ is undefined exactly when $\cos x = 0$, that is when

The domain of tangent is the set of all real numbers except these. At an excluded input $\tan x$ has no value at all. It is not equal to $\infty$.

Definition The zeros of tangent are the inputs where $\tan x = 0$. A quotient is zero exactly when its numerator is zero and its denominator is not, so the zeros are the zeros of sine,

At every one of these $\cos x = \pm 1 \ne 0$, so tangent is defined there.

Definition For every $x$ in the domain of tangent and every integer $k$, the input $x + k\pi$ is also in the domain and

The reason is that a half turn sends $P(x)$ to the opposite point $-P(x)$, which flips the sign of both coordinates and leaves their quotient unchanged. The least positive period of tangent is $\pi$. To see that nothing smaller works, suppose $p > 0$ is a period. Then $\tan p = \tan(0 + p) = \tan 0 = 0$. The zeros of tangent are exactly the inputs $k\pi$, so $p = k\pi$ for a positive integer $k$, and therefore $p \ge \pi$.

Definition The vertical line $x = a$ is a vertical asymptote of the graph of $f$ if, from at least one side of $a$, the values $f(x)$ become and stay larger than any number you name, or become and stay smaller than any negative number you name, once $x$ is close enough to $a$ on that side. In limit notation, at least one of the one-sided limits of $f$ at $a$ is $\infty$ or $-\infty$. Every excluded input $x = \frac{\pi}{2} + k\pi$ is a vertical asymptote of tangent. Near such an input the numerator $\sin x$ is close to $1$ or $-1$ while the denominator $\cos x$ is close to $0$, so

exceeds any bound once $x$ is close enough. On each side the sign of $\tan x$ does not change, and it is the sign of $\sin x$ divided by the sign of $\cos x$ there. For every asymptote, $\tan x$ is large and positive just to its left and large and negative just to its right.

Definition The graph of tangent is the set of points $(x, \tan x)$ for $x$ in its domain. It is made of identical branches, one on each open interval

On each such interval tangent is increasing, takes every real value once, and is separated from the next branch by a vertical asymptote. The range of tangent is all real numbers. Tangent is odd, with $\tan(-x) = -\tan x$ for every $x$ in its domain, since sine is odd and cosine is even, so the graph is symmetric about the origin. Tangent is increasing on each branch but not on its whole domain. For example $\tan \frac{\pi}{4} = 1$ while $\tan \frac{3\pi}{4} = -1$. The middle branch passes through

Definition The principal branch of tangent is its restriction to the open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. On this interval tangent is increasing, hence one-to-one, and its range is still all real numbers. This is the branch that is inverted to define the inverse tangent, so $\arctan y$ is the unique $x$ in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $\tan x = y$.