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 27

Capitolo 5, Problema 27

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/14 + i sin π/14)]⁷

Guida verificata passo dopo passo

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 $r^n (\cos (n\theta) + i \sin (n\theta))$.

Identify the given complex number's modulus and argument: here, $r = 12$ and $\theta = \frac{\pi}{14}$, and the power to raise it to is $n = 7$.

Apply DeMoivre's Theorem by raising the modulus to the 7th power: calculate $r^7 = 12^7$ (do not compute the exact value, just express it as \$12^7$).

Multiply the argument by 7 to find the new angle: $7 \times \frac{\pi}{14} = \frac{7\pi}{14} = \frac{\pi}{2}$.

Write the result in rectangular form using the cosine and sine of the new angle: $12^7 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$, then express $\cos \frac{\pi}{2}$ and $\sin \frac{\pi}{2}$ in their exact values to get the rectangular form.

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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 Angle Multiplication

When applying DeMoivre's Theorem, the angle θ is multiplied by n. Understanding how to compute cos(nθ) and sin(nθ) using trigonometric identities or formulas helps simplify the expression and convert it back to rectangular form.

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Double Angle Identities

Pratica correlata

Domanda del libro di testo

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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In Exercises 27–32, select the representations that do not change the location of the given point. (7, 140°) (−7, 320°)

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In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ

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In Exercises 27–32, select the representations that do not change the location of the given point. (4, 120°) (−4, 300°)

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In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)

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In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.

x = 2 sin t, y = 2 cos t; 0 ≤ t < 2π

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