Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
11. Graphing Complex Numbers
Powers of Complex Numbers (DeMoivre's Theorem)
6:42 minutes
Problem 60
Textbook Question
Textbook QuestionIn Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _ [√3 (cos 5π/18 + i sin 5π/18)]⁶
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Key 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 any complex number expressed in polar form as r(cos θ + i sin θ), the nth power of this complex number can be calculated as r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to powers and is essential for solving problems involving complex exponentiation.
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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 represented in two forms: rectangular form (a + bi, where a and b are real numbers) and polar form (r(cos θ + i sin θ), where r is the modulus and θ is the argument). Understanding how to convert between these forms is crucial for applying DeMoivre's Theorem and expressing the final answer correctly.
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Converting Complex Numbers from Polar to Rectangular Form
Trigonometric Functions
Trigonometric functions such as sine and cosine are fundamental in the context of complex numbers in polar form. They relate the angles and sides of triangles and are used in DeMoivre's Theorem to compute the cosine and sine of the angle multiplied by n. Familiarity with these functions is necessary for accurately calculating the powers of complex numbers.
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Introduction to Trigonometric Functions
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