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 5.2.53

Chapter 5, Problem 5.2.53

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

Verified step by step guidance

Identify the complex number in polar form: \(4(\cos 15^\circ + i \sin 15^\circ)\), where the modulus \(r = 4\) and the argument \(\theta = 15^\circ\).

Recall DeMoivre's Theorem, which states that for a complex number in polar form, \((r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\).

Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 4^3\) and the new argument as \(3 \times 15^\circ\).

Write the result in polar form: \(4^3 (\cos 45^\circ + i \sin 45^\circ)\).

Convert the polar form back to rectangular form using \(x = r^3 \cos 45^\circ\) and \(y = r^3 \sin 45^\circ\), 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, or in rectangular form as a + bi. Converting between these forms is essential for interpreting results after applying DeMoivre's Theorem.

Trigonometric Identities for Conversion

To convert from polar to rectangular form, use a = r cos θ and b = r sin θ. Understanding these identities helps express the final answer in a + bi form after applying DeMoivre's Theorem to powers of complex numbers.

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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 = 20(cos 75° + i sin 75°)

z₂ = 4(cos 25° + i sin 25°)

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. (√2 − i)⁴

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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. [√2 (cos (5π/6) + i sin (5π/6))]⁴

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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