Question: A civil engineer models the oscillation of a suspension bridge under wind load with the complex equation $(z - 3i)^8 = -16i$. Find the maximum imaginary part among all roots $z$, and express it in the form $\sin \theta$ for some $\theta \in (0, \pi)$.

["A Civil Engineer Models Wind-Induced Oscillations Using Complex Dynamics: Solving $(z - 3i)^8 = -16i$ to Find Maximum Imaginary Part", "In the design of suspension bridges, engineers often model dynamic responses under environmental loads such as wind. One such mathematical challenge arises when analyzing oscillatory behavior modeled using complex numbers. A representative model is given by the equation:", "$$\n(z - 3i)^8 = -16i\n$$", "This equation captures the amplification and phase shift of wind-induced vibrations across structural modes. Solving for $ z $, we determine the complex roots representing possible oscillation states, and among them, we identify the maximum imaginary part—critical for assessing resonance risks in real-world designs.", "We seek all complex solutions $ z $ to the equation and focus on the root with the largest imaginary component, then express it in the form $ \sin \ heta $ with $ \ heta \in (0, \pi) $, as required for normalized dynamic response analysis.", "---", "Step 1: Rewrite the Equation", "Let $ w = z - 3i $. Then the equation becomes:", "$$\nw^8 = -16i\n$$", "We solve for $ w $ by expressing $ -16i $ in polar form.", "Note that:\n- $ -16i = 16 \cdot (-i) $\n- $ -i = e^{-i\pi/2} $", "So:\n$$\n-16i = 16 \left( \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right) \right) = 16 e^{i(3\pi/2)}\n$$", "Using De Moivre’s Theorem, the eighth roots are given by:", "$$\nw_k = 16^{1/8} \cdot \left[ \cos\left( \frac{ \frac{3\pi}{2} + 2\pi k }{8} \right) + i \sin\left( \frac{3\pi}{2} + 2\pi k }{8} \right) \right], \quad k = 0, 1, \dots, 7\n$$", "Since $ 16^{1/8} = (2^4)^{1/8} = 2^{1/2} = \sqrt{2} $, we have:", "$$\nw_k = \sqrt{2} \left( \cos\left( \frac{3\pi + 4\pi k}{16} \right) + i \sin\left( \frac{3\pi + 4\pi k}{16} \right) \right)\n$$", "---", "Step 2: Express $ z $ in Terms of $ w $", "Recall $ z = w + 3i $. So:", "$$\nz_k = w_k + 3i = \sqrt{2} \cos\ heta_k + i\left( \sqrt{2} \sin\ heta_k + 3 \right)\n$$", "where $ \ heta_k = \dfrac{3\pi + 4\pi k}{16} = \dfrac{\pi(3 + 4k)}{16} $, for $ k = 0, 1, \dots, 7 $.", "Our goal is to find the maximum imaginary part among the $ z_k $, i.e., maximize:", "$$\n\ ext{Im}(z_k) = \sqrt{2} \sin\ heta_k + 3\n$$", "Since $ 3 $ is constant, maximizing the imaginary part reduces to maximizing $ \sin\ heta_k $ over $ k $.", "---", "Step 3: Maximize $ \sin\ heta_k $", "We compute $ \ heta_k = \dfrac{\pi(3 + 4k)}{16} $ for $ k = 0 $ to $ 7 $:", "| $ k $ | $ \ heta_k $ | $ \sin\ heta_k $ | $ \sqrt{2} \sin\ heta_k + 3 $ |\n|--------|----------------|--------------------|-------------------------------|\n| 0 | $ 3\pi/16 \approx 34.7^\circ $ | $ \approx 0.290 $ | $ \approx 3.414 $ |\n| 1 | $ 7\pi/16 \approx 78.75^\circ $ | $ \approx 0.980 $ | $ \approx 3.414 + 1.414 = 4.828 $? Wait—no: $ \sqrt{2} \cdot 0.980 \approx 1.386 \Rightarrow \approx 4.686 $ |\n| 2 | $ 11\pi/16 \approx 121.9^\circ $ | $ \sin(121.9^\circ) = \sin(58.1^\circ) \approx 0.850 $ | $ 1.386 + 3 = 4.386 $ |\n| 3 | $ 15\pi/16 \approx 168.75^\circ $ | $ \sin(168.75^\circ) \approx \sin(11.25^\circ) \approx 0.195 $ | $ \approx 1.386 + 0.585 = 1.971 $ |\n| 4 | $ 19\pi/16 \approx 213.75^\circ $ | $ \sin(\approx 213.75^\circ) = -\sin(33.75^\circ) \approx -0.556 $ | $ \approx -0.782 + 3 = 2.218 $ |\n| 5 | $ 23\pi/16 \approx 258.75^\circ $ | $ \sin(\approx 258.75^\circ) = -\sin(78.75^\circ) \approx -0.980 $ | $ \approx -1.386 + 3 = 1.614 $ |\n| 6 | $ 27\pi/16 \approx 303.75^\circ $ | $ \sin(\approx 303.75^\circ) = -\sin(56.25^\circ) \approx -0.832 $ | $ \approx -1.173 + 3 = 1.827 $ |\n| 7 | $ 31\pi/16 \approx 348.75^\circ $ | $ \sin(\approx 348.75^\circ) = -\sin(11.25^\circ) \approx -0.195 $ | $ \approx -0.277 + 3 = 2.723 $ |", "Wait—let’s compute more accurately using exact values.", "Note: $ \ heta_k = \dfrac{\pi(3 + 4k)}{16} $", "- $ k = 7 $: $ \ heta_7 = \frac{\pi(3 + 28)}{16} = \frac{31\pi}{16} = 2\pi - \frac{\pi}{16} $, so $ \sin\ heta_7 = \sin\left( -\frac{\pi}{16} \right) = -\sin\left( \frac{\pi}{16} \right) \approx -0.1951 $", "But we seek the maximum of $ \sin\ heta_k $. Among all $ k $, the largest sine occurs at $ k = 2 $:", "$$\n\ heta_2 = \frac{11\pi}{16} = \pi - \frac{5\pi}{16} \Rightarrow \sin\ heta_2 = \sin\left( \frac{5\pi}{16} \right) = \sin(56.25^\circ) \approx 0.8308\n$$", "Wait—earlier estimate was off. Let's compute precisely:", "- $ \frac{5\pi}{16} = 56.25^\circ $, $ \sin(56.25^\circ) \approx 0.8310 $\n- $ \frac{3\pi}{16} = 33.75^\circ $, $ \sin \approx 0.5556 $\n- $ \frac{7\pi}{16} = 78.75^\circ $, $ \sin \approx 0.9808 $\n- $ \frac{9\pi}{16} = 101.25^\circ $, $ \sin \approx 0.9816 $\n- $ \frac{11\pi}{16} = 121.875^\circ $, $ \sin \approx 0.9808 $\n- $ \frac{13\pi}{16} = 146.25^\circ $, $ \sin \approx 0.8310 $\n- $ \frac{15\pi}{16} = 168.75^\circ $, $ \sin \approx 0.1951 $", "Wait—$ \frac{7\pi}{16} = 78.75^\circ $, $ \sin(78.75^\circ) \approx 0.9808 $", "And $ k = 7 $: $ \ heta_7 = \frac{31\pi}{16} = 2\pi + \frac{31\pi}{16} - 2\pi = \frac{31\pi - 32\pi}{16} = -\frac{\pi}{16} $? No:", "$ \frac{31\pi}{16} = 1.9375\pi = 360^\circ \cdot \frac{31}{32} \approx 348.75^\circ $, so $ \sin(348.75^\circ) = -\sin(11.25^\circ) \approx -0.1951 $", "But $ k = 2 $: $ \ heta_2 = \frac{11\pi}{16} = 121.875^\circ $, $ \sin(121.875^\circ) = \sin(58.125^\circ) \approx \sin(58^\circ) \approx 0.829 $", "Wait—what about $ k = 1 $: $ \ heta_1 = \frac{7\pi}{16} = 78.75^\circ $, $ \sin \approx 0.9808 $", "And $ k = 2 $: $ \ heta_2 = \frac{11\pi}{16} = 121.875^\circ $, $ \sin(121.875^\circ) = \sin(58.125^\circ) \approx 0.829 $", "But $ k = 7 $: $ \ heta_7 = \frac{31\pi}{16} = 348.75^\circ $, $ \sin \approx -0.195 $", "Now check $ k = 6 $: $ \ heta_6 = \frac{27\pi}{16} = 303.75^\circ $, $ \sin(303.75^\circ) = -\sin(56.25^\circ) \approx -0.831 $", "But is there a $ k $ with $ \sin\ heta_k > \sin(7\pi/16) \approx 0.9808 $?", "Try $ k = 5 $: $ \frac{19\pi}{16} = 213.75^\circ $, $ \sin = -\sin(33.75^\circ) \approx -0.556 $", "No—maximum sine is at $ \ heta = \frac{7\pi}{16} \approx 78.75^\circ $, with $ \sin\ heta_2 \approx 0.9808 $", "But wait: $ k = 6 $: $ \ heta_6 = \frac{27\pi}{16} = \pi + \frac{11\pi}{16} = 180^\circ + 121.875^\circ = 301.875^\circ $, $ \sin \approx -\sin(58.125^\circ) \approx -0.831 $", "So maximum $ \sin\ heta_k $ occurs at $ k = 1 $: $ \ heta_1 = \frac{7\pi}{16} $, $ \sin\ heta_1 = \sin\left( \frac{7\pi}{16} \right) $", "Thus:", "$$\n\max \ ext{Im}(z_k) = \sqrt{2} \cdot \sin\left( \frac{7\pi}{16} \right) + 3\n$$", "But the problem asks to express the maximum imaginary part in the form $ \sin \ heta $, not $ \sqrt{2} \sin \ heta + 3 $. This suggests we may have misinterpreted.", "Wait — reconsider the root expression:", "Each $ z_k = \sqrt{2} \cos\ heta_k + i\left( \sqrt{2} \sin\ heta_k + 3 \right) $", "So $ \ ext{Im}(z_k) = \sqrt{2} \sin\ heta_k + 3 $", "We seek the maximum value of this.", "Since $ \sqrt{2} \sin\ heta_k \leq \sqrt{2} $, the maximum occurs when $ \sin\ heta_k $ is maximized among the roots, which is at $ k = 1 $, $ \ heta_k = \frac{7\pi}{16} $, so:", "$$\n\max \ ext{Im}(z_k) = \sqrt{2} \sin\left( \frac{7\pi}{16} \right) + 3\n$$", "But this is not of the form $ \sin \ heta $, unless we reframe.", "Wait — perhaps the question intends for us to extract the maximum value of the imaginary part and express it as $ \sin \ heta $, but scaled? No — the instruction says: “express it in the form $ \sin \ heta $”.", "This implies the maximum imaginary part itself is $ \sin \ heta $ for some $ \ heta \in (0, \pi) $. But $ \sqrt{2} \sin\left( \frac{7\pi}{16} \right) + 3 \approx 1.414 \cdot 0.9808 + 3 \approx 1.386 + 3 = 4.386 $, which exceeds 1 — impossible for a sine value.", "Ah — mistake: $ z_k $ is complex, and the imaginary part is $ \sqrt{2} \sin\ heta_k + 3 $, but this is not bounded by 1. However, in engineering contexts, relative amplitude is often normalized. But here, the equation $ (z - 3i)^8 = -16i $ suggests the physical amplitude scales with $ \sqrt{2} $, so the total imaginary projection can exceed 3.", "But the key insight: the maximum imaginary part among the roots is:", "$$\n\max \ ext{Im}(z_k) = \sqrt{2} \sin\left( \frac{7\pi}{16} \right) + 3\n$$", "But this is not $ \sin \ heta $. Re-examining the problem: “express it in the form $ \sin \ heta $” — likely a misstatement. But waiting — perhaps we are to find the maximum of the imaginary part of $ z $ and write it as $ \sin \ heta $ multiplied by a constant? No — the instruction says “express it in the form $ \sin \ heta $”.", "Alternative interpretation: perhaps the maximum value is $ \sin \ heta $ for some $ \ heta $? Only if normalized. But that contradicts magnitude.", "Wait — reconsider: in signal processing, such oscillatory modes are often rationalized. But here, the expression:", "$$\n\max \ ext{Im}(z_k) = \sqrt{2} \sin\left( "]