In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 10° + i sin 10°)]³

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Identify the complex number in polar form: \(2(\cos 10^\circ + i \sin 10^\circ)\), where the modulus \(r = 2\) and the argument \(\theta = 10^\circ\).

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 \(r^3 = 2^3\) and the new argument as \(3 \times 10^\circ\).

Write the result in polar form: \(2^3 (\cos 30^\circ + i \sin 30^\circ)\).

Convert the polar form back to rectangular form by calculating \(2^3 \cos 30^\circ\) for the real part and \(2^3 \sin 30^\circ\) for the imaginary part.

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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, where a and b are real numbers. Converting between these forms is essential for interpreting results.

Conversion from Polar to Rectangular Form

After applying DeMoivre's Theorem, the result is in polar form. To write the answer in rectangular form, use a = r cos θ and b = r sin θ to find the real and imaginary parts, respectively, expressing the complex number as a + bi.