In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 15° + i sin 15°)]³

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Identify the complex number in polar form: \(4(\cos 15^\circ + i \sin 15^\circ)\), where the modulus \(r = 4\) and the argument \(\theta = 15^\circ\).

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

Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 4^3\) and the new argument as \(3 \times 15^\circ\).

Write the result in polar form: \(4^3 (\cos 45^\circ + i \sin 45^\circ)\).

Convert the polar form back to rectangular form using \(x = r^3 \cos 45^\circ\) and \(y = r^3 \sin 45^\circ\), so the rectangular form is \(x + iy\).

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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.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ the argument, or in rectangular form as a + bi. Converting between these forms is essential for interpreting results after applying DeMoivre's Theorem.

Trigonometric Identities for Conversion

To convert from polar to rectangular form, use a = r cos θ and b = r sin θ. Understanding these identities helps express the final answer in a + bi form after applying DeMoivre's Theorem to powers of complex numbers.