Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 5.2.60

Chapter 5, Problem 5.2.60

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

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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 \(a + bi\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

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