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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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 81E
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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.
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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