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 27

Chapter 5, Problem 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

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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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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. Rectangular form is a + bi, where a and b are real numbers. Converting between these forms is essential for interpreting results.

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

Related Practice

Textbook Question

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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Textbook Question

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

Textbook Question

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

Textbook Question

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _ (1−i√3)²

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Textbook Question

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

Textbook Question

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