In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/12 + i sin π/12)]⁶

검증된 단계별 안내

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

Identify the given complex number's modulus and argument: here, \(r = \frac{1}{2}\) and \(\theta = \frac{\pi}{12}\).

Apply DeMoivre's Theorem to raise the complex number to the 6th power: compute \(r^6 = \left(\frac{1}{2}\right)^6\) and multiply the angle by 6 to get \(6 \times \frac{\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2}\).

Write the result in polar form: \(\left(\frac{1}{2}\right)^6 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)\).

Convert the polar form back to rectangular form by evaluating \(\cos \frac{\pi}{2}\) and \(\sin \frac{\pi}{2}\), then multiply by \(\left(\frac{1}{2}\right)^6\) to get the final rectangular form.

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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 rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting between these forms is essential for applying DeMoivre's Theorem and then rewriting the result back into rectangular form.

Trigonometric Identities for Cosine and Sine

Using trigonometric identities helps simplify expressions like cos(nθ) and sin(nθ) after applying DeMoivre's Theorem. Understanding these identities aids in accurately converting the final result into rectangular form.