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 5.2.62

Chapter 5, Problem 5.2.62

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

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Express the complex number \(1 - i\) in polar form. First, find the modulus \(r\) using \(r = \sqrt{a^2 + b^2}\), where \(a = 1\) and \(b = -1\).

Calculate the argument \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Remember to consider the correct quadrant for \(\theta\).

Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\).

Apply DeMoivre's Theorem to raise the complex number to the 5th power: \(\left[r(\cos \theta + i \sin \theta)\right]^5 = r^5 \left(\cos(5\theta) + i \sin(5\theta)\right)\).

Convert the result back to rectangular form by calculating \(r^5 \cos(5\theta)\) for the real part and \(r^5 \sin(5\theta)\) for the imaginary part.

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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 expressed in polar form as r(cos θ + i sin θ), its nth power is r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.

Conversion between Rectangular and Polar Forms

Complex numbers can be represented in rectangular form (a + bi) or polar form (r(cos θ + i sin θ)). Converting to polar form involves finding the magnitude r = √(a² + b²) and the argument θ = arctan(b/a), which is essential for applying DeMoivre's Theorem.

Rectangular Form of Complex Numbers

Rectangular form expresses complex numbers as a + bi, where a is the real part and b is the imaginary part. After using DeMoivre's Theorem in polar form, the result must be converted back to rectangular form by evaluating r^n cos nθ and r^n sin nθ to find the real and imaginary components.

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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 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. [√2 (cos (5π/6) + i sin (5π/6))]⁴

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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

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