Tangent and cotangent

Sine and cosine already name the two segments of a point on the unit circle. Once both segments are known, when is their quotient the object you need, rather than either segment alone?

The adjacent side stays 1

Freeze a right angle, and freeze the angle so that it is real and acute, $0<x<\pi /2$. Freeze the adjacent side at length $1$. The opposite side is the only length that still moves.

That opposite length is $\tan x$. If the opposite side has length $a$ and the adjacent side has length $b$,

$$\tan x=\frac{a}{b},\cot x=\frac{b}{a}.$$

Here $b=1$, so $a$ itself is the tangent. On this triangle the cotangent is only the reciprocal of that ratio. The next picture draws it as a length. Sine and cosine already write the other two ratios, opposite over the hypotenuse and adjacent over the hypotenuse, under the same right angle.

The same triangle cuts the unit circle

Keep the adjacent side equal to $1$, and put the unit circle inside that triangle. The angle is measured in radians. The ray from the origin meets the vertical line through $(1,0)$ at height $\tan x$, and it meets the horizontal line through $(0,1)$ at width $\cot x$. The vertical side you just dragged is that first cut.

Past a right angle the ray turns back

Keep the radius equal to $1$ and let the angle pass $\pi /2$. The forward ray still meets the horizontal line, now to the left of $(0,1)$, so $\cot x$ is negative. The vertical cut lies on the backward extension, below $(1,0)$, so $\tan x$ is negative.

The quotient, off those freezes

The pictures keep a real angle and a unit radius. For every complex $x$ other than $x=\pi /2+k\pi$, with $k$ an integer,

$$\tan x:=\frac{\sin x}{\cos x},$$

and for every complex $x$ other than $x=k\pi$,

$$\cot x:=\frac{\cos x}{\sin x}.$$

Where both are defined, each is the reciprocal of the other.

A right-angle shift

Keep the angle real and acute. The horizontal cut of that angle has the same length as the vertical cut of the complementary angle.

For every complex $x$ where both sides are defined,

$$\cot x=\tan \left(\frac{\pi }{2}-x\right).$$

Every identity for cotangent is the matching identity for tangent after that shift.

The same angles, already tabulated

Where both entries are numbers, each is the reciprocal of the other. Those exact values sit on the sine and cosine page. This page reads them as the side ratio above and as the two cuts.

Tangent and cotangent are the quotients of sine and cosine. With a right angle and an adjacent side of length $1$ they are that side ratio, and on the unit circle they are two cuts. The next questions are where the quotients meet zero, where they blow up, and how the graphs repeat.