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)
6:53 minutes
Problem 61
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. (1 + i)⁵
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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 the 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 Form of Complex Numbers
The polar form of a complex number is a way of expressing it using its magnitude (r) and angle (θ) instead of its rectangular coordinates (a + bi). The magnitude is calculated as r = √(a² + b²), and the angle is found using θ = arctan(b/a). This representation is crucial for applying DeMoivre's Theorem effectively.
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Complex Numbers In Polar Form
Rectangular Form of Complex Numbers
The rectangular form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. Converting the result from polar form back to rectangular form involves using the cosine and sine values to find the real and imaginary components, which is necessary for presenting the final answer in the required format.
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Converting Complex Numbers from Polar to Rectangular Form
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Master Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem) with a bite sized video explanation from Nick Kaneko
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