Question: Compute $ \left( \cos 45^\circ + i \sin 45^\circ \right)^8 $ using De Moivre’s Theorem.
["# Compute $ \left( \cos 45^\circ + i \sin 45^\circ \right)^8 $ Using De Moivre’s Theorem", "When working with complex numbers in polar form, De Moivre’s Theorem provides a powerful shortcut for raising expressions of the form $ \cos \ heta + i \sin \ heta $ to powers. This article explores how to compute $ \left( \cos 45^\circ + i \sin 45^\circ \right)^8 $ efficiently, using De Moivre’s Theorem and fundamental identities from trigonometry and complex analysis.", "## Understanding De Moivre’s Theorem", "De Moivre’s Theorem states that for any real number $ \ heta $ and integer $ n $,\n$$\n\left( \cos \ heta + i \sin \ heta \right)^n = \cos (n\ heta) + i \sin (n\ heta).\n$$\nThis elegant formula simplifies exponentiation of complex numbers written in polar form, turning multiplicative operations into straightforward angular additions.", "## Apply the Theorem to the Given Expression", "Let’s define:\n$$\nz = \cos 45^\circ + i \sin 45^\circ.\n$$\nBy De Moivre’s Theorem, raising $ z $ to the 8th power yields:\n$$\nz^8 = \left( \cos 45^\circ + i \sin 45^\circ \right)^8 = \cos (8 \ imes 45^\circ) + i \sin (8 \ imes 45^\circ).\n$$", "Calculate the angle:\n$$\n8 \ imes 45^\circ = 360^\circ.\n$$", "## Simplify Using Trigonometric Identities", "Now evaluate:\n$$\n\cos 360^\circ + i \sin 360^\circ.\n$$\nRecall that:\n- $ \cos 360^\circ = \cos 0^\circ = 1 $,\n- $ \sin 360^\circ = \sin 0^\circ = 0 $.", "Therefore,\n$$\n\cos 360^\circ + i \sin 360^\circ = 1 + i \cdot 0 = 1.\n$$", "## Conclusion", "Thus, using De Moivre’s Theorem and fundamental trigonometric values:\n$$\n\left( \cos 45^\circ + i \sin 45^\circ \right)^8 = 1.\n$$", "This result demonstrates the power of polar form and De Moivre’s Theorem in simplifying complex exponentiation, especially with angles related to standard positions on the unit circle.", "Whether solving problems in electrical engineering, physics, or pure mathematics, mastering this technique enables faster, more intuitive calculations involving complex numbers.", "---", "Keywords: De Moivre’s Theorem, complex numbers, $ \cos 45^\circ + i \sin 45^\circ $, exponentiation, trigonometry, $ \cos(n\ heta) + i \sin(n\ heta) $, polar form, $ 1 $ result, mathematical identity."]
