In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶

검증된 단계별 안내

Identify the complex number in polar form: \(z = r(\cos \theta + i \sin \theta)\), where \(r = \sqrt{3}\) and \(\theta = \frac{5\pi}{18}\).

Recall DeMoivre's Theorem, which states that for any integer \(n\), \(z^n = r^n (\cos(n\theta) + i \sin(n\theta))\).

Apply DeMoivre's Theorem with \(n = 6\): compute \(z^6 = (\sqrt{3})^6 \left( \cos \left(6 \times \frac{5\pi}{18} \right) + i \sin \left(6 \times \frac{5\pi}{18} \right) \right)\).

Simplify the magnitude: calculate \((\sqrt{3})^6\) by expressing it as \((3^{1/2})^6 = 3^{3}\).

Simplify the angle: multiply \(6 \times \frac{5\pi}{18}\) to find the new angle, then use the values of \(\cos\) and \(\sin\) at this angle to write the answer in rectangular form \(a + bi\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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, simplifying calculations.

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)). Converting between these forms is essential, especially after applying DeMoivre's Theorem, to write the final answer in rectangular form as required.

Trigonometric Identities and Angle Multiplication

When applying DeMoivre's Theorem, the angle θ is multiplied by the power n. Understanding trigonometric identities and how to simplify angles (e.g., using periodicity or sum/difference formulas) helps in accurately finding cos(nθ) and sin(nθ) for the final expression.