ANSWER
[tex]- \frac{\pi}{4} [/tex]
EXPLANATION
The given function is,
[tex]f(t) = 2 \csc(t + \frac{\pi}{4} ) - 1[/tex]
This function has a period of
[tex]2\pi[/tex]
just as the parent function
[tex]f(t) = \csc(t ) [/tex]
A sample period of this parent function is
[tex][ 0 , 2\pi][/tex]
Which begins at zero.
For the transformed function,
[tex]f(t) = 2 \csc(t + \frac{\pi}{4} ) - 1[/tex]
There has been a horizontal shift of
[tex] \frac{\pi}{4} [/tex]
to the left.
The transformed function will have a sample period,
[tex][ - \frac{\pi}{4} , \frac{7\pi}{4} ][/tex]
Therefore a sample period begins at
[tex]- \frac{\pi}{4} [/tex]
