Question

Graph of 1/sinx.

When x=0, the function goes to infinity and the function is even.

Is it correct to say that since it is periodic, the graph of 1/sinx will be the copy of the function, but in the other direction? Is it correct for all periodic functions?

Answer 1

• no
• no

Explanation:

See the attached for a graph of 1/sin(x) = csc(x). Just as sin(x) is an odd function, so is 1/sin(x).

Any odd function is symmetrical about the origin, so the left side of the graph is a copy of the right side, rotated 180° about the origin ("in the other direction"). That is, csc(x) = -csc(-x). This is true because the function is odd, not because it is periodic.

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Csc(x) is also a periodic function with a period of 2π. That means ...

csc(x+2π) ≡ csc(x)

This sort of replication of the function is true for all periodic functions (where the added value, the horizontal translation, is the period of the function).

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You will note that the relation applicable to an odd function (-f(-x) = f(x)) is different from the relation applicable to a periodic function (f(x) = f(x+period)). An odd periodic function, such as csc(x), will be described by both relations.