Textbook Question
In Exercises 32–35, find all the complex roots. Write roots in rectangular form.The complex fourth roots of 16 (cos 2π/3 + i sin 2π/3)
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11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
3:39 minutesProblem 7
Textbook Question
In Exercises 5–7, perform the indicated operation. Leave answers in polar form.[2(cos 10° + i sin 10°)]⁵
Verified step by step guidance1
Recognize that the expression is in polar form, specifically in the form of $r(\cos \theta + i \sin \theta)$, where $r = 2$ and $\theta = 10^\circ$.
Apply De Moivre's Theorem, which states that for a complex number in polar form $r(\cos \theta + i \sin \theta)$, the $n$-th power is given by $r^n(\cos(n\theta) + i \sin(n\theta))$.
Substitute $r = 2$, $\theta = 10^\circ$, and $n = 5$ into De Moivre's Theorem to get $2^5(\cos(5 \times 10^\circ) + i \sin(5 \times 10^\circ))$.
Calculate $2^5$ to find the new magnitude of the complex number.
Multiply $5$ by $10^\circ$ to find the new angle in the polar form, which is $50^\circ$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Form of Complex Numbers
Polar form expresses complex numbers in terms of their magnitude and angle, represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument. This form is particularly useful for multiplication and exponentiation of complex numbers, as it simplifies calculations by allowing the use of trigonometric identities.
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De Moivre's Theorem
De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth power can be calculated as r^n(cos(nθ) + i sin(nθ)). This theorem is essential for raising complex numbers to a power, as it provides a straightforward method to compute the resulting magnitude and angle.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Key identities, such as the sine and cosine addition formulas, are often used in conjunction with polar forms to simplify expressions and solve problems involving angles, especially when performing operations like addition or multiplication of complex numbers.
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