6. \( \frac{3\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)}{2\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)} \)

To simplify the expression \( \frac{3\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)}{2\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)} \), we can utilize Euler's formula which states that \( \cos \theta + i \sin \theta = e^{i\theta} \). Thus, we rewrite the expression as: \[ \frac{3 e^{i 200^{\circ}}}{2 e^{i 50^{\circ}}} \] This can be simplified by dividing the exponential terms: \[ \frac{3}{2} e^{i (200^{\circ} - 50^{\circ})} = \frac{3}{2} e^{i 150^{\circ}} \] Next, we convert back into rectangular form using Euler's formula: \[ e^{i 150^{\circ}} = \cos 150^{\circ} + i \sin 150^{\circ} \] Calculating \( \cos 150^{\circ} \) and \( \sin 150^{\circ} \): \[ \cos 150^{\circ} = -\frac{\sqrt{3}}{2}, \quad \sin 150^{\circ} = \frac{1}{2} \] Now substituting these back into our expression gives us: \[ \frac{3}{2} \left(-\frac{\sqrt{3}}{2} + i \frac{1}{2}\right) = \frac{3}{2} \left(-\frac{\sqrt{3}}{2}\right) + \frac{3}{2} \left(i \frac{1}{2}\right) \] \[ = -\frac{3\sqrt{3}}{4} + i \frac{3}{4} \] Thus, the final simplified form of the original expression is: \[ -\frac{3\sqrt{3}}{4} + i \frac{3}{4} \]