Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 5.2.56

Capitolo 5, Problema 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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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.

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Percorso guidato

03:41

Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)

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.

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03:58

Converting Complex Numbers from Polar to Rectangular Form

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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05:06

Double Angle Identities

Pratica correlata

Domanda del libro di testo

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

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Domanda del libro di testo

In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 4 csc θ

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Domanda del libro di testo

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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Domanda del libro di testo

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7

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In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/12 + i sin π/12)]⁶

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Domanda del libro di testo

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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