An online STEM student is exploring complex numbers and discovers the expression \((\cos rac{\pi}{6} + i \sin rac{\pi}{6})^6\). Compute this value and express it in the form \(a + bi\).
["Exploring Complex Numbers: Evaluating ((\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})^6)", "Complex numbers play a crucial role in advanced mathematics, engineering, and physics, and Euler’s Formula offers a powerful way to work with them. One fascinating expression students often encounter is:", "[\n(\cos \ heta + i \sin \ heta)^n\n]", "where (\ heta) is an angle in radians and (n) is an integer. This form relates directly to De Moivre’s Theorem, which states:", "[\n(\cos \ heta + i \sin \ heta)^n = \cos(n\ heta) + i \sin(n\ heta)\n]", "Today, we explore a specific example relevant to online STEM learners studying complex analysis: evaluating\n[\n\left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)^6\n]", "### Step 1: Apply De Moivre’s Theorem", "Using De Moivre’s Theorem, we compute:", "[\n\left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)^6 = \cos\left(6 \cdot \frac{\pi}{6}\right) + i \sin\left(6 \cdot \frac{\pi}{6}\right)\n]", "Simplifying the angle:", "[\n6 \cdot \frac{\pi}{6} = \pi\n]", "So the expression becomes:", "[\n\cos \pi + i \sin \pi\n]", "### Step 2: Evaluate Trigonometric Functions", "We know from standard trigonometric values:", "[\n\cos \pi = -1, \quad \sin \pi = 0\n]", "Thus:", "[\n\cos \pi + i \sin \pi = -1 + i \cdot 0 = -1\n]", "### Final Answer", "[\n\left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)^6 = -1\n]", "This result is neatly expressed in the form (a + bi) as:", "[\na = -1, \quad b = 0 \quad \Rightarrow \quad -1 + 0i\n]", "### Conclusion", "This example beautifully illustrates how complex numbers in polar form simplify powerful computations using trigonometric identities. Students studying STEM online can deepen their understanding by applying De Moivre’s Theorem to evaluate such powers efficiently. Mastering these concepts opens doors to deeper insights in engineering, signal processing, quantum mechanics, and beyond.", "Keywords: complex numbers, De Moivre’s Theorem, Poisson’s formula, (\cos \ heta + i \sin \ heta), CAS, STEM learning, (\left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)^6), (-1 + 0i), advanced mathematics."]
