3. (a) Using Euler's formula, derive the identities 3 cos³(z) = cos(z)+cos (37), sin³ (x)=sin(x)-sin(32). (b) Use the result of part (a) to find a general solution of y" + 4y = cos(r).
3. (a) Using Euler's formula, derive the identities 3 cos³(z) = cos(z)+cos (37), sin³ (x)=sin(x)-sin(32). (b) Use the result of part (a) to find a general solution of y" + 4y = cos(r).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.2: Trigonometric Functions Of Angles
Problem 83E
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![### Educational Content on Euler’s Formula and Trigonometric Identities
#### Problem Statement
**3.** (a) Using Euler's formula, derive the identities
\[ \cos^3(x) = \frac{3}{4} \cos(x) + \frac{1}{4} \cos(3x), \]
\[ \sin^3(x) = \frac{3}{4} \sin(x) - \frac{1}{4} \sin(3x). \]
(b) Use the result of part (a) to find a general solution of the differential equation
\[ y'' + 4y = \cos^3(x). \]
---
#### Explanation
**Euler’s Formula:**
Euler's formula states that for any real number \( x \),
\[ e^{ix} = \cos(x) + i\sin(x). \]
This formula can be used to express trigonometric identities and solve complex problems involving trigonometric functions.
**Trigonometric Identities Derivation:**
- In part (a), you are expected to derive the identities for \(\cos^3(x)\) and \(\sin^3(x)\) using Euler's formula. This involves expressing powers of sine and cosine in terms of fundamental sine and cosine functions with different arguments (e.g., \(3x\)).
- The provided identities show how a power of cosine or sine can be written as a weighted sum of cosines or sines with multiple angles.
**Differential Equation:**
- In part (b), utilize the derived trigonometric identities to find a general solution of the given differential equation \(y'' + 4y = \cos^3(x)\). This step likely involves using the method of undetermined coefficients or another similar technique to solve non-homogeneous linear differential equations. The derived identity for \(\cos^3(x)\) serves as the non-homogeneous part of the equation, simplifying the problem-solving process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F590efbb4-bfa3-48e4-b7a4-ba8d0f00ec23%2Ff8340ea9-0969-4a72-b49e-dd5687d64ec1%2F958wei_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content on Euler’s Formula and Trigonometric Identities
#### Problem Statement
**3.** (a) Using Euler's formula, derive the identities
\[ \cos^3(x) = \frac{3}{4} \cos(x) + \frac{1}{4} \cos(3x), \]
\[ \sin^3(x) = \frac{3}{4} \sin(x) - \frac{1}{4} \sin(3x). \]
(b) Use the result of part (a) to find a general solution of the differential equation
\[ y'' + 4y = \cos^3(x). \]
---
#### Explanation
**Euler’s Formula:**
Euler's formula states that for any real number \( x \),
\[ e^{ix} = \cos(x) + i\sin(x). \]
This formula can be used to express trigonometric identities and solve complex problems involving trigonometric functions.
**Trigonometric Identities Derivation:**
- In part (a), you are expected to derive the identities for \(\cos^3(x)\) and \(\sin^3(x)\) using Euler's formula. This involves expressing powers of sine and cosine in terms of fundamental sine and cosine functions with different arguments (e.g., \(3x\)).
- The provided identities show how a power of cosine or sine can be written as a weighted sum of cosines or sines with multiple angles.
**Differential Equation:**
- In part (b), utilize the derived trigonometric identities to find a general solution of the given differential equation \(y'' + 4y = \cos^3(x)\). This step likely involves using the method of undetermined coefficients or another similar technique to solve non-homogeneous linear differential equations. The derived identity for \(\cos^3(x)\) serves as the non-homogeneous part of the equation, simplifying the problem-solving process.
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