Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem 3: Trigonometric Equation
**Task:**
Solve the equation: \( 4 \sin x + 3 \cos x = 5 \)
In this exercise, you are required to solve the trigonometric equation for the variable \( x \). Here are the detailed steps to guide you through the process:
### Solution Approach:
1. **Recognize the type of equation:**
The given equation is a linear combination of sine and cosine functions.
2. **Rewrite the equation using the trigonometric identity:**
Consider an angle \(\theta\) such that \( \cos \theta = \frac{a}{R} \) and \( \sin \theta = \frac{b}{R} \) where \( R = \sqrt{a^2 + b^2} \). For our equation \( 4 \sin x + 3 \cos x = 5 \):
Let \( R = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).
Therefore, the equation can be rewritten as:
\[
4 \sin x + 3 \cos x = R (\cos \theta \sin x + \sin \theta \cos x) = R \sin(x + \theta)
\]
Here, \(\theta\) is such that \( \cos \theta = \frac{3}{5} \) and \( \sin \theta = \frac{4}{5} \).
3. **Simplify and solve for \( x \):**
Substitute back into the original equation:
\[
5 \sin(x + \theta) = 5
\]
Simplify:
\[
\sin(x + \theta) = 1
\]
Therefore:
\[
x + \theta = \frac{\pi}{2} + 2k\pi \quad \text{for integer } k
\]
4. **Find the specific solutions for \( x \):**
Given \( \theta = \arctan \frac{4}{3}\), calculate \(\theta\):
\[
\theta \approx 0.927 \text{ radians}
\]
Hence:
\[
x = \frac{\pi}{2}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedc716be-e066-4234-bd1a-1a86837575c3%2F67503a3c-9234-4305-b522-8da38962bd13%2Fwp6133g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3: Trigonometric Equation
**Task:**
Solve the equation: \( 4 \sin x + 3 \cos x = 5 \)
In this exercise, you are required to solve the trigonometric equation for the variable \( x \). Here are the detailed steps to guide you through the process:
### Solution Approach:
1. **Recognize the type of equation:**
The given equation is a linear combination of sine and cosine functions.
2. **Rewrite the equation using the trigonometric identity:**
Consider an angle \(\theta\) such that \( \cos \theta = \frac{a}{R} \) and \( \sin \theta = \frac{b}{R} \) where \( R = \sqrt{a^2 + b^2} \). For our equation \( 4 \sin x + 3 \cos x = 5 \):
Let \( R = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\).
Therefore, the equation can be rewritten as:
\[
4 \sin x + 3 \cos x = R (\cos \theta \sin x + \sin \theta \cos x) = R \sin(x + \theta)
\]
Here, \(\theta\) is such that \( \cos \theta = \frac{3}{5} \) and \( \sin \theta = \frac{4}{5} \).
3. **Simplify and solve for \( x \):**
Substitute back into the original equation:
\[
5 \sin(x + \theta) = 5
\]
Simplify:
\[
\sin(x + \theta) = 1
\]
Therefore:
\[
x + \theta = \frac{\pi}{2} + 2k\pi \quad \text{for integer } k
\]
4. **Find the specific solutions for \( x \):**
Given \( \theta = \arctan \frac{4}{3}\), calculate \(\theta\):
\[
\theta \approx 0.927 \text{ radians}
\]
Hence:
\[
x = \frac{\pi}{2}
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