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 1
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11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
5:58 minutesProblem 27
Textbook Question
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/14 + i sin π/14)]⁷
Verified step by step guidance1
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 \(r^n (\cos (n\theta) + i \sin (n\theta))\).
Identify the given complex number's modulus and argument: here, \(r = 12\) and \(\theta = \frac{\pi}{14}\), and the power to raise it to is \(n = 7\).
Apply DeMoivre's Theorem by raising the modulus to the 7th power: calculate \(r^7 = 12^7\) (do not compute the exact value, just express it as \$12^7$).
Multiply the argument by 7 to find the new angle: \(7 \times \frac{\pi}{14} = \frac{7\pi}{14} = \frac{\pi}{2}\).
Write the result in rectangular form using the cosine and sine of the new angle: \(12^7 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)\), then express \(\cos \frac{\pi}{2}\) and \(\sin \frac{\pi}{2}\) in their exact values to get the rectangular form.
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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. Rectangular form is a + bi, where a and b are real numbers. Converting between these forms is essential for interpreting results.
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Trigonometric Identities for Angle Multiplication
When applying DeMoivre's Theorem, the angle θ is multiplied by n. Understanding how to compute cos(nθ) and sin(nθ) using trigonometric identities or formulas helps simplify the expression and convert it back to rectangular form.
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