Textbook Question
In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64
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11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
5:55 minutesProblem 26
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
Verified step by step guidance1
Identify the complex number in polar form: \(4(\cos 50^\circ + i \sin 50^\circ)\), where the modulus \(r = 4\) and the argument \(\theta = 50^\circ\).
Recall DeMoivre's Theorem, which states that for a complex number in polar form, \((r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\).
Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 4^3\) and the new argument as \(3 \times 50^\circ\).
Write the result in polar form: \(4^3 (\cos 150^\circ + i \sin 150^\circ)\).
Convert the polar form back to rectangular form by calculating \(4^3 \cos 150^\circ\) for the real part and \(4^3 \sin 150^\circ\) for the imaginary part.
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5mKey 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.
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Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)
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, or in rectangular form as a + bi. Converting between these forms is essential for interpreting and simplifying results.
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Conversion from Polar to Rectangular Form
After applying DeMoivre's Theorem, the result is in polar form. To write the answer in rectangular form, use a = r cos θ and b = r sin θ to find the real and imaginary parts, respectively, expressing the complex number as a + bi.
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