Write the first 4 terms of the Maclaurin series for .cose Write the first 4 terms of the Maclaurin series for sine Use the above Maciaurin series expansions of e".cosd,and sind to justify Euler's Formula =Cose + isine. Why are we able to rearrange terms in this equality?
Write the first 4 terms of the Maclaurin series for .cose Write the first 4 terms of the Maclaurin series for sine Use the above Maciaurin series expansions of e".cosd,and sind to justify Euler's Formula =Cose + isine. Why are we able to rearrange terms in this equality?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Exploring Maclaurin Series and Euler's Formula
#### Maclaurin Series for \( \cos(\theta) \)
Write the first 4 terms of the Maclaurin series for \( \cos(\theta) \).
#### Maclaurin Series for \( \sin(\theta) \)
Write the first 4 terms of the Maclaurin series for \( \sin(\theta) \).
#### Connection to Euler's Formula
Use the above Maclaurin series expansions of \( e^{i\theta} \), \( \cos(\theta) \), and \( \sin(\theta) \) to justify Euler's Formula:
\[ e^{i\theta} = \cos(\theta) + i\sin(\theta). \]
Why are we able to rearrange terms in this equality?
---
The above instructions prompt students to:
1. **Expand the Maclaurin Series**:
- For \( \cos(\theta) \):
\[
\cos(\theta) = 1 - \frac{\theta^2}{2} + \frac{\theta^4}{24} - \frac{\theta^6}{720} + \cdots
\]
- For \( \sin(\theta) \):
\[
\sin(\theta) = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \frac{\theta^7}{5040} + \cdots
\]
2. **Apply the Series to Euler's Formula**:
- Using the series expansions of \( e^{i\theta} \), \( \cos(\theta) \), and \( \sin(\theta) \):
\[
e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2} - i\frac{\theta^3}{6} + \frac{\theta^4}{24} + i\frac{\theta^5}{120} - \cdots
\]
Coupled with the trigonometric series, students are to demonstrate how these expansions support the famous identity by comparing real and imaginary parts.
3. **Explore Term Rearrangement**:
- Discuss why the terms can be rearranged when equating \( e^{i\theta} \) to \( \cos(\theta) + i\sin(\theta) \). This usually hinges on the concept of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4f75436-93ee-4e58-a0e3-afe0fa0e1ea5%2Fdf2ba734-9851-4820-9306-93973f32f876%2F9inakct_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exploring Maclaurin Series and Euler's Formula
#### Maclaurin Series for \( \cos(\theta) \)
Write the first 4 terms of the Maclaurin series for \( \cos(\theta) \).
#### Maclaurin Series for \( \sin(\theta) \)
Write the first 4 terms of the Maclaurin series for \( \sin(\theta) \).
#### Connection to Euler's Formula
Use the above Maclaurin series expansions of \( e^{i\theta} \), \( \cos(\theta) \), and \( \sin(\theta) \) to justify Euler's Formula:
\[ e^{i\theta} = \cos(\theta) + i\sin(\theta). \]
Why are we able to rearrange terms in this equality?
---
The above instructions prompt students to:
1. **Expand the Maclaurin Series**:
- For \( \cos(\theta) \):
\[
\cos(\theta) = 1 - \frac{\theta^2}{2} + \frac{\theta^4}{24} - \frac{\theta^6}{720} + \cdots
\]
- For \( \sin(\theta) \):
\[
\sin(\theta) = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \frac{\theta^7}{5040} + \cdots
\]
2. **Apply the Series to Euler's Formula**:
- Using the series expansions of \( e^{i\theta} \), \( \cos(\theta) \), and \( \sin(\theta) \):
\[
e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2} - i\frac{\theta^3}{6} + \frac{\theta^4}{24} + i\frac{\theta^5}{120} - \cdots
\]
Coupled with the trigonometric series, students are to demonstrate how these expansions support the famous identity by comparing real and imaginary parts.
3. **Explore Term Rearrangement**:
- Discuss why the terms can be rearranged when equating \( e^{i\theta} \) to \( \cos(\theta) + i\sin(\theta) \). This usually hinges on the concept of
![**Topic: Exploring Exponents and the Maclaurin Series Expansion**
**Using properties of exponents, complete the following:**
\[
i^0 = \_\_\_\_ \quad
i^1 = \_\_\_\_ \quad
i^2 = \_\_\_\_ \quad
i^3 = \_\_\_\_
\]
\[
i^4 = \_\_\_\_ \quad
i^5 = \_\_\_\_ \quad
i^6 = \_\_\_\_ \quad
i^7 = \_\_\_\_
\]
**Task:**
Use the Maclaurin series for \(e^x\) and write the first 9 terms of the expansion of \(e^{ix}\) and simplify powers of \(i\).
---
**Instructional Notes:**
Students should use their knowledge of the imaginary unit \(i\), where \(i \) is defined as the square root of \(-1\). Recall that:
- \(i^0 = 1\)
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\) (repeats every 4 powers)
Based on these properties, complete the values for higher powers.
The Maclaurin series expansion for \(e^x\) is given by:
\[
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
\]
For \(e^{ix}\), substitute \(ix\) for \(x\):
\[
e^{ix} = \sum_{n=0}^\infty \frac{(ix)^n}{n!}
\]
Simplify this expansion by separating real and imaginary parts and compute the first 9 terms. Use the properties of \(i\) to simplify the resulting expression.
---
This exercise will help in understanding the cyclical nature of powers of \(i\) and connect exponential functions with complex numbers through series expansions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4f75436-93ee-4e58-a0e3-afe0fa0e1ea5%2Fdf2ba734-9851-4820-9306-93973f32f876%2F8c6gi4y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Exploring Exponents and the Maclaurin Series Expansion**
**Using properties of exponents, complete the following:**
\[
i^0 = \_\_\_\_ \quad
i^1 = \_\_\_\_ \quad
i^2 = \_\_\_\_ \quad
i^3 = \_\_\_\_
\]
\[
i^4 = \_\_\_\_ \quad
i^5 = \_\_\_\_ \quad
i^6 = \_\_\_\_ \quad
i^7 = \_\_\_\_
\]
**Task:**
Use the Maclaurin series for \(e^x\) and write the first 9 terms of the expansion of \(e^{ix}\) and simplify powers of \(i\).
---
**Instructional Notes:**
Students should use their knowledge of the imaginary unit \(i\), where \(i \) is defined as the square root of \(-1\). Recall that:
- \(i^0 = 1\)
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\) (repeats every 4 powers)
Based on these properties, complete the values for higher powers.
The Maclaurin series expansion for \(e^x\) is given by:
\[
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
\]
For \(e^{ix}\), substitute \(ix\) for \(x\):
\[
e^{ix} = \sum_{n=0}^\infty \frac{(ix)^n}{n!}
\]
Simplify this expansion by separating real and imaginary parts and compute the first 9 terms. Use the properties of \(i\) to simplify the resulting expression.
---
This exercise will help in understanding the cyclical nature of powers of \(i\) and connect exponential functions with complex numbers through series expansions.
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