A periodic function #f# with period #rho# has the property:
#f(x+rho)=f(x)#
Generally, when we say #rho#, we mean #nrho_0#, for #n in ZZ#, where #rho_0# is the principal period of the function and is defined as the smallest positive period of #f#.
For the sine function and many more trigonometric functions, the principal period is #rho_0 = 2pi#. You can see this by the Unit Circle; when you add #2pi# to whatever angle you had, it just loops around and falls back on the same position.
Thus,
#sin(x+2pi)=sinx#
And finally, since the set of periods is #rho = nrho_0#, we have
#sin(x+color(red)(2npi))=sinx#, for any integer #n#.
In our case, you can see that #8pi = 2*4pi#, so the following relation is true:
#color(blue)(sin(theta+8pi)=sintheta#
