1. **State the problem:** We have two complex numbers represented as vectors in the plane: - $z_2$ with magnitude 8 and angle $\theta$ from the negative x-axis (second quadrant). - $z_1$ with magnitude 4 and angle $\theta$ from the positive x-axis (fourth quadrant). We want to find the ratio $\frac{z_1}{z_2}$. 2. **Express the complex numbers in polar form:** - Since $z_2$ is in the second quadrant with angle $\theta$ from the negative x-axis, its angle from the positive x-axis is $\pi - \theta$. So, $$z_2 = 8(\cos(\pi - \theta) + i\sin(\pi - \theta))$$ - Since $z_1$ is in the fourth quadrant with angle $\theta$ from the positive x-axis, its angle is $-\theta$ (measured clockwise). So, $$z_1 = 4(\cos(-\theta) + i\sin(-\theta))$$ 3. **Formula for division of complex numbers in polar form:** $$\frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\alpha_1 - \alpha_2) + i \sin(\alpha_1 - \alpha_2) \right)$$ where $r_1, r_2$ are magnitudes and $\alpha_1, \alpha_2$ are arguments (angles) of $z_1$ and $z_2$ respectively. 4. **Calculate the magnitude ratio:** $$\frac{r_1}{r_2} = \frac{4}{8} = \frac{\cancel{4}}{\cancel{8}} = \frac{1}{2}$$ 5. **Calculate the angle difference:** $$\alpha_1 - \alpha_2 = (-\theta) - (\pi - \theta) = -\theta - \pi + \theta = -\pi$$ 6. **Write the ratio:** $$\frac{z_1}{z_2} = \frac{1}{2} \left( \cos(-\pi) + i \sin(-\pi) \right)$$ 7. **Evaluate trigonometric functions:** - $\cos(-\pi) = \cos(\pi) = -1$ - $\sin(-\pi) = -\sin(\pi) = 0$ So, $$\frac{z_1}{z_2} = \frac{1}{2}(-1 + 0i) = -\frac{1}{2}$$ **Final answer:** $$\boxed{\frac{z_1}{z_2} = -\frac{1}{2}}$$