Solution: Apply De Moivre’s theorem: $(\cos 15^\circ + i \sin 15^\circ)^{12} = \cos(180^\circ) + i \sin(180^\circ) = -1 + 0i$.
["Apply De Moivre’s Theorem: Simplify $(\cos 15^\circ + i \sin 15^\circ)^{12}$ Easily", "Mathematics often reveals elegant shortcuts through powerful theorems, and De Moivre’s Theorem is one such gem—especially when working with complex numbers in polar form. In this article, we explore how applying De Moivre’s Theorem simplifies the expression $(\cos 15^\circ + i \sin 15^\circ)^{12}$ and clearly demonstrates that this equals $-1 + 0i$.", "---", "### What is De Moivre’s Theorem?", "De Moivre’s Theorem states that for any real number $\ heta$ and integer $n$:", "$$\n(\cos \ heta + i \sin \ heta)^n = \cos(n\ heta) + i \sin(n\ heta)\n$$", "This powerful identity transforms complex numbers expressed in trigonometric polar form into simple trigonometric expressions raised to a power—making exponentiation much easier.", "---", "### Applying De Moivre’s Theorem to the Given Expression", "Consider the expression:\n$$\n(\cos 15^\circ + i \sin 15^\circ)^{12}\n$$", "Applying De Moivre’s Theorem:", "$$\n(\cos 15^\circ + i \sin 15^\circ)^{12} = \cos(12 \ imes 15^\circ) + i \sin(12 \ imes 15^\circ)\n$$", "Calculate the angle:\n$$\n12 \ imes 15^\circ = 180^\circ\n$$", "So the expression becomes:\n$$\n\cos(180^\circ) + i \sin(180^\circ)\n$$", "We know from trigonometry:\n$$\n\cos(180^\circ) = -1, \quad \sin(180^\circ) = 0\n$$", "Therefore:\n$$\n\cos(180^\circ) + i \sin(180^\circ) = -1 + 0i\n$$", "---", "### Why This Simplification Matters (SEO Keywords: complex numbers computation, De Moivre’s Theorem explanation, trigonometric identities)", "This computation demonstrates how De Moivre’s Theorem drastically simplifies calculating powers of complex numbers. Instead of multiplying $(\cos 15^\circ + i \sin 15^\circ)$ twelve times, we:\n- Multiply the angle: $15^\circ \ imes 12 = 180^\circ$\n- Retain the same trigonometric form with the new angle\n- Evaluate the simplified sine and cosine values instantly", "This method is especially useful for engineers, physicists, and students solving problems involving rotations, waves, or oscillations where complex exponentials are common.", "---", "### Final Result", "$$\n(\cos 15^\circ + i \sin 15^\circ)^{12} = \cos(180^\circ) + i \sin(180^\circ) = -1 + 0i\n$$", "This elegant result proves the power of De Moivre’s Theorem in reducing complex exponentiation to straightforward trigonometric evaluation.", "---", "### TL;DR", "- Use De Moivre’s Theorem: $(\cos \ heta + i \sin \ heta)^n = \cos(n\ heta) + i \sin(n\ heta)$\n- Compute powers of complex numbers in polar form efficiently\n- Example: $(\cos 15^\circ + i \sin 15^\circ)^{12} = \cos(180^\circ) + i \sin(180^\circ) = -1$\n- A key tool for simplifying trigonometric expressions and complex exponentiation", "---", "Keywords: De Moivre’s Theorem, complex numbers, trigonometric identities, exponential form, polar coordinates, $ \cos 15^\circ + i \sin 15^\circ $, $ (\cos \ heta + i \sin \ heta)^n $, $ \cos(180^\circ) + i \sin(180^\circ) $, $-1 + 0i$", "Make complex math simpler—apply De Moivre’s Theorem today!"]
