Solution: By De Moivre’s theorem, $ (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) $. Applying $ n = 5 $, the result is $ \cos(5\theta) + i \sin(5\theta) $. \boxed{\cos(5\theta) + i \sin(5\theta)}

["Mastering Complex Numbers: De Moivre’s Theorem Simplified with n = 5", "When exploring the fascinating world of complex numbers, De Moivre’s Theorem stands out as a powerful and elegant tool. This theorem provides a straightforward way to raise complex numbers expressed in polar form to integer powers—transforming trigonometric expressions with remarkable clarity.", "## What Is De Moivre’s Theorem?", "De Moivre’s Theorem states that for any real angle ( \ heta ) and integer ( n ):", "[\n(\cos \ heta + i \sin \ heta)^n = \cos(n\ heta) + i \sin(n\ heta)\n]", "This formula reveals how multiplying a complex number on the unit circle by itself repeatedly (i.e., raising it to a power) corresponds to multiplying its angle by that exponent—while preserving its magnitude.", "## Applying n = 5: Unlocking $ \cos(5\ heta) + i \sin(5\ heta) $", "Let’s apply De Moivre’s Theorem with ( n = 5 ). Starting from the polar representation:", "[\n(\cos \ heta + i \sin \ heta)^5 = \cos(5\ heta) + i \sin(5\ heta)\n]", "This result dramatically simplifies what would otherwise be a lengthy trigonometric expansion. Instead of expanding ( (\cos \ heta + i \sin \ heta)^5 ) using binomial theorem and lengthy identities, we directly obtain the result using the theorem.", "### The Expanded Form (Optional Insight)", "While we often prefer the compact polar form, the expansion confirms its correctness:", "[\n(\cos \ heta + i \sin \ heta)^5 = \sum_{k=0}^{5} \binom{5}{k} (\cos \ heta)^{5-k} (i \sin \ heta)^k\n]", "Manipulating each term using ( i^2 = -1 ) and trigonometric identities eventually yields:", "[\n\cos(5\ heta) + i \sin(5\ heta)\n]", "This match proves the theorem’s validity, especially when ( n = 5 ), a commonly used integer value in mathematical problems.", "## Why Is This Useful?", "- Simplifies Powers: Raising complex numbers to powers becomes algebraic, avoiding messy expansions.\n- Facilitates Trigonometric Identities: Deriving multiple-angle formulas becomes intuitive using De Moivre’s result.\n- Applies Across Fields: Used in signal processing, quantum mechanics, and electrical engineering for analyzing oscillations and waves.", "## Conclusion", "De Moivre’s Theorem is more than a formula—it’s a bridge between geometry and algebra in the complex plane. With ( n = 5 ), we obtain a clean expression:", "[\n\boxed{\cos(5\ heta) + i \sin(5\ heta)}\n]", "This result exemplifies the theorem’s elegance and power. Whether for solving equations, modeling waves, or exploring symmetries, mastering De Moivre’s identity is essential for anyone advancing in mathematics, physics, or engineering.", "Keywords: De Moivre’s theorem, complex numbers, $ (\cos \ heta + i \sin \ heta)^n $, $ \cos(5\ heta) + i \sin(5\ heta) $, complex exponentiation, trigonometric identities, polar form, mathematical theorem explained."]