Finding rules for reciprocal circular functions?

[Yoda]

How do you find the rule for reciprocal circular functions from a graph when you're given asymptotes.
For example, the graph y=cosec(a(x-b)) has asymptotes at x=-pi/4, pi/4, 3pi/4 and 5pi/4. Find the values of a and b.

[e^1]

As cosec is the reciprocal of the sin function, notice that sin(0)=0 which would make csc(0) undefined. The first positive x-asymptote is x=pi/4. Thus we can make b = pi/4, as a horizontal translation of the cosec function.

You can use the period of the cosec function to find the value of b:



For a cosec function, there are three asymptotes from the start to the end of a period (from observation). On another note, sin(x) crosses the x-axis 3 times in a single period.

Using this observation, the period equals 5pi/4 - pi/4 = pi. Now pi = 2pi/a  => a = 2.

a = 2, b = pi/4

(Sorry if there's no latex, doesn't seem to be working right now)