In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 50° + i sin 50°)]³

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Identify the complex number in polar form: \(4(\cos 50^\circ + i \sin 50^\circ)\), where the modulus \(r = 4\) and the argument \(\theta = 50^\circ\).

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 \(3 \times 50^\circ\).

Write the result in polar form: \(4^3 (\cos 150^\circ + i \sin 150^\circ)\).

Convert the polar form back to rectangular form by calculating \(4^3 \cos 150^\circ\) for the real part and \(4^3 \sin 150^\circ\) for the imaginary part.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 and simplifying 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.