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 5.2.60

Capitolo 5, Problema 5.2.60

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶

Guida verificata passo dopo passo

Identify the complex number in polar form: $z = r(\cos \theta + i \sin \theta)$, where $r = \sqrt{3}$ and $\theta = \frac{5\pi}{18}$.

Recall DeMoivre's Theorem, which states that for any integer $n$, $z^n = r^n (\cos(n\theta) + i \sin(n\theta))$.

Apply DeMoivre's Theorem with $n = 6$: compute $z^6 = (\sqrt{3})^6 \left( \cos \left(6 \times \frac{5\pi}{18} \right) + i \sin \left(6 \times \frac{5\pi}{18} \right) \right)$.

Simplify the magnitude: calculate $(\sqrt{3})^6$ by expressing it as $(3^{1/2})^6 = 3^{3}$.

Simplify the angle: multiply $6 \times \frac{5\pi}{18}$ to find the new angle, then use the values of $\cos$ and $\sin$ at this angle to write the answer in rectangular form $a + bi$.

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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, simplifying calculations.

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

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Converting Complex Numbers from Polar to Rectangular Form

Trigonometric Identities and Angle Multiplication

When applying DeMoivre's Theorem, the angle θ is multiplied by the power n. Understanding trigonometric identities and how to simplify angles (e.g., using periodicity or sum/difference formulas) helps in accurately finding cos(nθ) and sin(nθ) for the final expression.

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05:06

Double Angle Identities

Pratica correlata

Domanda del libro di testo

In Exercises 61–63, test for symmetry with respect to

a. the polar axis.

b. the line θ = π/2.

c. the pole.

r = 5 + 3 cos θ

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In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range. x = t² + t + 1, y = 2t

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In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2

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In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. θ = 3π/4

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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. [2(cos 10° + i sin 10°)]³

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In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. r = 5 csc θ

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