The remaining problems require you to construct a mathematical proof by induction. Remember, if you don't make use of the inductive hypothesis, there is no way to finish the proof correctly! 9. The following formula is called De'Moivre's Formula: (cos+ i sin 0)" = cos(ne)+ i sin(ne) Prove De'Moivre's formula for all integers n ≥0. (Hint: you will need a trig identity from Precalculus).
The remaining problems require you to construct a mathematical proof by induction. Remember, if you don't make use of the inductive hypothesis, there is no way to finish the proof correctly! 9. The following formula is called De'Moivre's Formula: (cos+ i sin 0)" = cos(ne)+ i sin(ne) Prove De'Moivre's formula for all integers n ≥0. (Hint: you will need a trig identity from Precalculus).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Inductive Proof Exercise: DeMoivre's Formula
**Instructions for Inductive Proof:**
The remaining problems require you to construct a mathematical proof by induction. Remember, if you don’t make use of the inductive hypothesis, there is no way to finish the proof correctly!
#### Problem 9
The following formula is called DeMoivre’s Formula:
\[
\left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta)
\]
**Task**: Prove DeMoivre’s formula for all integers \( n \geq 0 \).
**Hint**: You will need a trig identity from Precalculus.
---
**Explanation:**
In this problem, you are asked to establish the truth of DeMoivre’s formula using mathematical induction. The formula relates powers of complex numbers in polar form to their corresponding trigonometric identities, a fundamental result in complex number theory.
**Steps for Induction**:
1. **Base Case**: Verify the formula for \( n = 0 \).
2. **Inductive Step**: Assume the formula holds for some arbitrary \( n = k \). Show it holds for \( n = k + 1 \) using your assumption.
Mathematical proofs by induction often require recalling foundational trigonometric identities, so it’s beneficial to review those, especially double angle formulas and other relationships that tie into Euler’s formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c836437-a2a4-49ad-8637-6abf33735fd4%2F9beefaae-0cc4-42fe-a6f0-da25b9dfb4c0%2Feq3cgik_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Inductive Proof Exercise: DeMoivre's Formula
**Instructions for Inductive Proof:**
The remaining problems require you to construct a mathematical proof by induction. Remember, if you don’t make use of the inductive hypothesis, there is no way to finish the proof correctly!
#### Problem 9
The following formula is called DeMoivre’s Formula:
\[
\left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta)
\]
**Task**: Prove DeMoivre’s formula for all integers \( n \geq 0 \).
**Hint**: You will need a trig identity from Precalculus.
---
**Explanation:**
In this problem, you are asked to establish the truth of DeMoivre’s formula using mathematical induction. The formula relates powers of complex numbers in polar form to their corresponding trigonometric identities, a fundamental result in complex number theory.
**Steps for Induction**:
1. **Base Case**: Verify the formula for \( n = 0 \).
2. **Inductive Step**: Assume the formula holds for some arbitrary \( n = k \). Show it holds for \( n = k + 1 \) using your assumption.
Mathematical proofs by induction often require recalling foundational trigonometric identities, so it’s beneficial to review those, especially double angle formulas and other relationships that tie into Euler’s formula.
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