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 25

Chapter 5, Problem 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

Verified step by step guidance

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 \(a + bi\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 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.

Recommended video:

5:32

Fundamental Trigonometric 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°)

views

Textbook Question

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.

z₁ = 5 (cos 4π/3 + i sin 4π/3)

z₂ = 10 (cos π/3 + i sin π/3)

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 = √t, y = t − 1

views

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 21–28, divide and express the result in standard form. 8i / 4−3i

views

Textbook Question

In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (4, π/2)