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 5.2.59

Chapter 5, Problem 5.2.59

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√2 (cos (5π/6) + i sin (5π/6))]⁴

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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, \(r = \sqrt{2}\) and \(\theta = \frac{5\pi}{6}\).

Apply DeMoivre's Theorem for \(n=4\): compute the new modulus as \(r^4 = (\sqrt{2})^4\) and the new argument as \(4 \times \frac{5\pi}{6}\).

Write the resulting complex number in trigonometric form: \(r^4 \left( \cos \left(4 \times \frac{5\pi}{6} \right) + i \sin \left(4 \times \frac{5\pi}{6} \right) \right)\).

Convert the trigonometric form to rectangular form by calculating \(\cos\) and \(\sin\) of the new angle and multiplying each by \(r^4\) to get the real and imaginary parts respectively.

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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 in trigonometric form.

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)). Polar form uses magnitude r and angle θ, which is useful for multiplication and powers. Converting back to rectangular form involves using a = r cos θ and b = r sin θ.

Trigonometric Identities for Angle Multiplication

When applying DeMoivre's Theorem, the angle θ is multiplied by n. Understanding trigonometric identities and periodicity helps simplify cos(nθ) and sin(nθ) to find exact values, especially when angles are multiples of π, ensuring accurate conversion back to rectangular form.

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Related Practice

Textbook Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)

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Textbook Question

In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.

z₁ = 20(cos 75° + i sin 75°)

z₂ = 4(cos 25° + i sin 25°)

Textbook Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 15° + i sin 15°)]³

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Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x = 7

Textbook Question

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)

Textbook Question

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