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Graph of #2csc(2x - 1)#

graph{2 csc(2x - 1) [-10, 10, -5, 5]}

The trig functions like # csc x # all have period #2\pi.# By doubling the coefficient on #x#, that halves the period, so the function #csc(2x)# must have a period of #pi#, as must #2 csc(2x-1)#.

The phase shift for #csc(ax-b)# is given by #b/a.# Here we have a phase shift of #frac 1 2# radian, approximately #28.6^\circ#. The minus sign means #2csc(2x-1)# leads #2csc(2x)# so we call this a positive phase shift of #frac 1 2# radian.

#csc(x) = 1/sin(x)# so it diverges twice per period. The amplitude is infinite.