1. Stating the problem: Evaluate the expression $$\arctan \left[ \frac{(1+4 \times 3218)^{1/3}}{\sin (\Re (e^{i \pi}))} \right] / \pi$$ 2. Calculate the numerator inside the cube root: $$1 + 4 \times 3218 = 1 + 12872 = 12873$$ 3. Compute the cube root: $$12873^{1/3}$$ This is the cube root of 12873, which we will keep as $12873^{1/3}$ for now. 4. Evaluate the denominator: First, evaluate $e^{i \pi}$ using Euler's formula: $$e^{i \pi} = \cos \pi + i \sin \pi = -1 + 0i$$ The real part $\Re(e^{i \pi})$ is: $$\Re(e^{i \pi}) = -1$$ 5. Compute $\sin(\Re(e^{i \pi})) = \sin(-1)$: Since sine is an odd function: $$\sin(-1) = -\sin(1)$$ 6. Substitute these results back: $$\arctan \left[ \frac{12873^{1/3}}{-\sin(1)} \right] / \pi = \arctan \left[ -\frac{12873^{1/3}}{\sin(1)} \right] / \pi$$ 7. Estimate numerical values: $$12873^{1/3} \approx 23.0$$ $$\sin(1) \approx 0.84147$$ Thus, $$-\frac{12873^{1/3}}{\sin(1)} \approx -\frac{23.0}{0.84147} \approx -27.32$$ 8. Calculate $\arctan(-27.32)$: For large negative values, $\arctan(x) \to -\frac{\pi}{2}$, So, $$\arctan(-27.32) \approx -\frac{\pi}{2}$$ 9. Divide by $\pi$: $$\frac{-\frac{\pi}{2}}{\pi} = -\frac{1}{2}$$ Final answer: $$\boxed{-\frac{1}{2}}$$