Need understanding how to evaluate a trigonometric function using its period as an aid.

Keeping in mind that, if we convert the "improper" π fraction to a mixed π fraction, we'd get



now, check the picture below, the angle is just a full revolution, "clockwise", and then π/3 extra, that's the angel on the IV quadrant.

using the period I assume it means the cycle, and you can just check your Unit Circle to see what the sine of that angle is.