In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√2 (cos (5π/6) + i sin (5π/6))]⁴

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Recall DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)th power is given by \(r^n (\cos (n\theta) + i \sin (n\theta))\).

Identify the given complex number's modulus and argument: here, \(r = \sqrt{2}\) and \(\theta = \frac{5\pi}{6}\).

Apply DeMoivre's Theorem for \(n=4\): compute the new modulus as \(r^4 = (\sqrt{2})^4\) and the new argument as \(4 \times \frac{5\pi}{6}\).

Write the resulting complex number in trigonometric form: \(r^4 \left( \cos \left(4 \times \frac{5\pi}{6} \right) + i \sin \left(4 \times \frac{5\pi}{6} \right) \right)\).

Convert the trigonometric form to rectangular form by calculating \(\cos\) and \(\sin\) of the new angle and multiplying each by \(r^4\) to get the real and imaginary parts respectively.

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주요 개념

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DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It allows raising complex numbers to integer powers by multiplying the angle and raising the magnitude to the power, simplifying calculations in trigonometric form.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Polar form uses magnitude r and angle θ, which is useful for multiplication and powers. Converting back to rectangular form involves using a = r cos θ and b = r sin θ.

Trigonometric Identities for Angle Multiplication

When applying DeMoivre's Theorem, the angle θ is multiplied by n. Understanding trigonometric identities and periodicity helps simplify cos(nθ) and sin(nθ) to find exact values, especially when angles are multiples of π, ensuring accurate conversion back to rectangular form.