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 5.2.56

Chapter 5, Problem 5.2.56

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 40° + i sin 40°)]³

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Identify the complex number in polar form: \(2(\cos 40^\circ + i \sin 40^\circ)\), where the modulus \(r = 2\) and the argument \(\theta = 40^\circ\).

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 \(3 \times 40^\circ\).

Write the resulting complex number in polar form: \(2^3 (\cos 120^\circ + i \sin 120^\circ)\).

Convert the polar form back to rectangular form by calculating \(2^3 \cos 120^\circ\) for the real part and \(2^3 \sin 120^\circ\) for the imaginary part, then express the answer as \(a + bi\).

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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.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ) or in rectangular form as a + bi. Converting between these forms is essential, especially after applying DeMoivre's Theorem, to write the final answer in standard a + bi format.

Trigonometric Identities for Powers and Angles

Understanding how to compute cos(nθ) and sin(nθ) for multiples of angles is crucial. This involves using angle multiplication and sometimes trigonometric identities to simplify expressions when applying DeMoivre's Theorem.

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Double Angle Identities

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°

Textbook Question

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

√(−8) (√(−3) − √5 )

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Textbook Question

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)

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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

Textbook Question

In Exercises 21–28, divide and express the result in standard form.

2+3i / 2+i

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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.

Ellipse: Center: (−2,3); Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center

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