Simplify \( \sin \left(240^{\circ}\right) \) without the use of a calculator. Write your answer in simplest surd form if necessary.

To simplify \( \sin(240^{\circ}) \), we can first recognize that \( 240^{\circ} \) is in the third quadrant. The sine function is negative in the third quadrant. We can find the reference angle by calculating \( 240^{\circ} - 180^{\circ} = 60^{\circ} \). Using the sine value of the reference angle, we have: \[ \sin(240^{\circ}) = -\sin(60^{\circ}) \] We know from trigonometric values that \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \). Therefore, \[ \sin(240^{\circ}) = -\frac{\sqrt{3}}{2} \] Thus, the simplified form of \( \sin(240^{\circ}) \) is: \[ \boxed{-\frac{\sqrt{3}}{2}} \]