In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 − i)⁵

검증된 단계별 안내

Express the complex number \(1 - i\) in polar form. First, find the modulus \(r\) using \(r = \sqrt{a^2 + b^2}\), where \(a = 1\) and \(b = -1\).

Calculate the argument \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Remember to consider the correct quadrant for \(\theta\).

Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\).

Apply DeMoivre's Theorem to raise the complex number to the 5th power: \(\left[r(\cos \theta + i \sin \theta)\right]^5 = r^5 \left(\cos(5\theta) + i \sin(5\theta)\right)\).

Convert the result back to rectangular form by calculating \(r^5 \cos(5\theta)\) for the real part and \(r^5 \sin(5\theta)\) 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 expressed in polar form as r(cos θ + i sin θ), its nth power is r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.

Conversion between Rectangular and Polar Forms

Complex numbers can be represented in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting to polar form involves finding the magnitude r = √(a² + b²) and the argument θ = arctan(b/a), which is essential for applying DeMoivre's Theorem.

Rectangular Form of Complex Numbers

Rectangular form expresses complex numbers as a + bi, where a is the real part and b is the imaginary part. After using DeMoivre's Theorem in polar form, the result must be converted back to rectangular form by evaluating r^n cos nθ and r^n sin nθ to find the real and imaginary components.