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 29

Chapter 5, Problem 29

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (−2 − 2i)⁵

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Express the complex number \((-2 - 2i)\) in polar form. First, find the modulus \(r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4}\).

Next, find the argument \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right) = \tan^{-1}\left(\frac{-2}{-2}\right)\). Adjust \(\theta\) to the correct quadrant since both parts are negative.

Apply DeMoivre's Theorem, which states that for a complex number in polar form \(r(\cos \theta + i \sin \theta)\), its \(n\)-th power is \(r^n (\cos(n\theta) + i \sin(n\theta))\). Here, \(n = 5\).

Calculate \(r^5\) and multiply the argument by 5 to get \(5\theta\). Then write the expression \(r^5 (\cos(5\theta) + i \sin(5\theta))\).

Convert the result back to rectangular form by evaluating \(r^5 \cos(5\theta)\) for the real part and \(r^5 \sin(5\theta)\) for the imaginary part.

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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 expressed in polar form as r(cos θ + i sin θ), its nth power is given by r^n (cos nθ + i sin nθ). This theorem simplifies raising complex numbers to powers by working with their magnitude and angle instead of expanding binomials.

Conversion Between Rectangular and Polar Forms

To apply DeMoivre's Theorem, a complex number must be converted from rectangular form (a + bi) to polar form (r, θ), where r is the magnitude √(a² + b²) and θ is the argument arctan(b/a). This conversion is essential for using trigonometric identities in complex number powers.

Rectangular Form of Complex Numbers

After applying DeMoivre's Theorem in polar form, the result is converted back to rectangular form a + bi by calculating a = r^n cos(nθ) and b = r^n sin(nθ). Writing the answer in rectangular form makes it easier to interpret and use in further calculations.

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