Textbook Question
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 50° + i sin 50°)]³
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11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
7:28 minutesProblem 31
Textbook Question
In Exercises 30–31, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 125(cos 165° + i sin 165°)
Verified step by step guidance1
Identify the given complex number in polar form: \(125(\cos 165^\circ + i \sin 165^\circ)\), where the modulus \(r = 125\) and the argument \(\theta = 165^\circ\).
Recall that to find the cube roots of a complex number in polar form \(r(\cos \theta + i \sin \theta)\), the roots are given by: \(\sqrt[3]{r} \left( \cos \left( \frac{\theta + 360^\circ k}{3} \right) + i \sin \left( \frac{\theta + 360^\circ k}{3} \right) \right)\), where \(k = 0, 1, 2\).
Calculate the cube root of the modulus: \(\sqrt[3]{125}\), which will be the modulus of each root.
For each integer \(k = 0, 1, 2\), compute the argument of each root by evaluating \(\frac{165^\circ + 360^\circ k}{3}\).
Write each root in polar form as \(r_k (\cos \theta_k + i \sin \theta_k)\) using the modulus from step 3 and the arguments from step 4.
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7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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, the nth roots can be found by 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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Finding Complex Roots
To find the nth roots of a complex 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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