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

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Blitzer 3rd Edition

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

Problema 25

Capitolo 5, Problema 25

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 20° + i sin 20°)]³

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Identify the complex number in trigonometric form: $2(\cos 20^\circ + i \sin 20^\circ)$, where the modulus $r = 2$ and the argument $\theta = 20^\circ$.

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 $3 \times 20^\circ$.

Write the resulting complex number in trigonometric form: $2^3 (\cos 60^\circ + i \sin 60^\circ)$.

Convert the trigonometric form to rectangular form by calculating $2^3 \cos 60^\circ$ for the real part and $2^3 \sin 60^\circ$ 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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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), 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 is essential for interpreting results.

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Converting Complex Numbers from Polar to Rectangular Form

Trigonometric Identities for Conversion

To convert from polar to rectangular form after applying DeMoivre's Theorem, use trigonometric functions: a = r cos θ and b = r sin θ. Understanding sine and cosine values for given angles is crucial for accurate conversion.

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Fundamental Trigonometric Identities

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