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 5.2.55

Chapter 5, Problem 5.2.55

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 80° + i sin 80°)]³

Verified step by step guidance

Identify the complex number in polar form: \(2(\cos 80^\circ + i \sin 80^\circ)\), where the modulus \(r = 2\) and the argument \(\theta = 80^\circ\).

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 \(r^n (\cos n\theta + i \sin n\theta)\).

Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 2^3\) and the new argument as \(3 \times 80^\circ\).

Write the resulting complex number in polar form: \(2^3 (\cos 240^\circ + i \sin 240^\circ)\).

Convert the polar form back to rectangular form using \(x = r^3 \cos 240^\circ\) and \(y = r^3 \sin 240^\circ\), so the rectangular form is \(x + iy\).

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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.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ the argument. Rectangular form is a + bi, where a and b are real numbers. Converting between these forms involves trigonometric functions.

Conversion from Polar to Rectangular Form

To convert a complex number from polar to rectangular form, use a = r cos θ and b = r sin θ. This step is essential after applying DeMoivre's Theorem to express the result as a + bi, which is the standard rectangular form.

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Converting Complex Numbers from Polar to Rectangular Form

Related Practice

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 = cos 80° + i sin 80°

z₂ = cos 200° + i sin 200°

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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.

x² = 6y

292

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Textbook Question

In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)

921

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Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form.

(3√(−5) )( −4√(−12) )

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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. 3x + y = 7

726

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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.

Circle: Center: (3,5); Radius: 6

676

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