Graphing Cotangent Functions

Graphing Cotangent Functions

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

Graph

y = cot x = 1/tan x

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 = :
when tan x = 0.

Example

Sketch the given function's graph. y = cot (x - pi/2) (0 <= x <= 2pi)

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.