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.
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 :


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

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