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 5.2.65

Capitolo 5, Problema 5.2.65

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 9(cos 30° + i sin 30°)

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Identify the given complex number in polar form: \(9(\cos 30^\circ + i \sin 30^\circ)\). Here, the modulus \(r = 9\) and the argument \(\theta = 30^\circ\).

Recall that to find the complex square roots of a number in polar form \(r(\cos \theta + i \sin \theta)\), we use the formula for the \(n\)th roots: \(\sqrt[n]{r} \left( \cos \frac{\theta + 360^\circ k}{n} + i \sin \frac{\theta + 360^\circ k}{n} \right)\), where \(k = 0, 1, ..., n-1\). Since we want square roots, \(n=2\).

Calculate the modulus of the roots by taking the square root of \(r\): \(\sqrt{9} = 3\).

Calculate the arguments of the roots by dividing the original argument plus \(360^\circ k\) by 2 for \(k=0\) and \(k=1\): \(\frac{30^\circ + 360^\circ \times 0}{2}\) and \(\frac{30^\circ + 360^\circ \times 1}{2}\).

Write the two roots in polar form as \(3 \left( \cos \alpha + i \sin \alpha \right)\) where \(\alpha\) are the two arguments found in the previous step.

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Complex Numbers in Polar Form

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form simplifies multiplication, division, and finding roots by working with magnitudes and angles separately.

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Complex Numbers In Polar Form

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, raising it to the power n results in r^n (cos nθ + i sin nθ). Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, adding multiples of 360°/n for all roots.

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Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)

Finding Complex Roots

To find the complex nth roots of a number, calculate the nth root of the magnitude and determine the arguments by dividing the original angle by n and adding k(360°/n) for k = 0, 1, ..., n-1. This yields all distinct roots evenly spaced around the circle.

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Complex Roots

Pratica correlata

Domanda del libro di testo

In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.

z₁ = cos 80° + i sin 80°

z₂ = cos 200° + i sin 200°

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In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.

x² = 6y

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In Exercises 37–52, perform the indicated operations and write the result in standard form.

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In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 + i)² − (3 − i)²

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In Exercises 37–52, perform the indicated operations and write the result in standard form.

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In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Line: Passes through (−2,4) and (1,7)

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