# Cotangent Graph

Cotangent graph is opposite to that of tangent graph.This graph is also discontinuous graph because we know that

Cot x = cos x / sin x

For some values of x, sin x has 0 value.

For example,

**x = ..., -3π, -2π, -π, 0, π, 2π, 3π, 4π, 5π, ...**

for these values the denominator becomes zero and the cot x is undefined at these particular value. So there is a gap and discontinuity.

The periodic function of cot x is π This means it repeats itself after each π as we go left to right on the graph.

So considering the values of cos x and sin x for different values of x we can sketch the graph of y = cot x as follows.

**Note that there are vertical asymptotes (the blue dotted lines) where the denominator of cot x has value zero.**

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As there is a phase shift in the sine and cosine graph, in the same way there is a phase shift in cotangent graph.

y = a cot(bx + c)

bx + c = 0

⇒ x = -c/b which is the first cycle.

Period = π/|b|

For every cycle add k(π/|b|) that gives you the asymptotes.

**Example: y = cot(4x - π/2)**

4x - π/2 = 0

⇒ x = π/8

Period = π/b = π/8

As there is 4x so divide the cycle in 4 pieces of length and as there is negative sign, the phase shift is at left.

It will start from π/8 = 2π/16 and then add π/16 to each cycle.

It will end at π/8 + 4π/16= 6π/16.

So the graph will be between 2π/16 to 6π/16. So it will look like as follows :

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**Practice**

**1) y = cot(2x).**

a) Find the period.

b) Find the equation of vertical asymptote.

**2) y = cot(3x - π/2)**

a) Find the period.

b) Phase shift.

c) Graph the given function.

**Cotangent graph**

Graph Dictionary

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