# Graphing Cotangent Functions

How to graph cotangent functions: graph, example, and its solution.

## Graph

Cotangent is the reciprocal of tangent:*y* = cot *x* = 1/tan *x*.

See the graphs of*y* = tan *x* (gray) and*y* = cot *x* (white, blue).

If *y* = tan *x* = 0,

then those *x* values

are the vertical asymptotes of *y* = cot *x*.

(*y* = cot *x* = 1/[±0] = ±∞)

If *y* = tan *x* → ±∞,

then *y* = cot *x* = 0.

(*y* = cot *x* = 1/[±∞] = ±0 = 0)

The intersection of *y* = tan *x* and *y* = cot *x* is *y* = ±1.

(±1 = 1/[±1] = ±1)

So, to graph the cotangent function,

first draw the tangent function,

then draw the cotangent function.

Graphing tangent functions

The blue part is one cycle of *y* = cot *x*.

So the period of *y* = cot *x* is *π*:

same as *y* = tan *x*.

The asymptotes are *x* = *nπ*:

when tan *x* = 0.

## Example

Roughly draw one cycle of *y* = tan (*x* - *π*/2)

with its details.

Graphing tangent functions*y* = tan (*x* - *π*/2) is under the translation

(*x*, *y*) → (*x* + *π*/2, *y*).

So the central point is (*π*/2, 0).

Translation of a function

The period is *π* / |1| = *π*.

So the half-period is *π*/2.

So the left asymptote is *x* = *π*/2 - *π*/2 = 0

And the right asymptote is *x* = *π*/2 + *π*/2 = *π*.

Use these values to draw*y* = tan (*x* - *π*/2). (gray)

Then draw *y* = cot (*x* - *π*/2)

by using the tangent graph.

If *y* = tan (*x* - *π*/2) = 0,

then those *x* values

are the vertical asymptotes of *y* = cot (*x* - *π*/2).

If *y* = tan (*x* - *π*/2) → ±∞,

then *y* = cot (*x* - *π*/2) = 0.