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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![**Task: Convert a Trigonometric Expression**
Rewrite the expression \(-4 \sin(x) - 5 \cos(x)\) in the form \(A \sin(x + \phi)\).
**Solution Steps:**
1. **Amplitude Calculation (\(A\))**
Use the formula:
\[
A = \sqrt{a^2 + b^2}
\]
where \(a = -4\) and \(b = -5\).
2. **Phase Shift Calculation (\(\phi\))**
Use the arctangent to find \(\phi\):
\[
\phi = \tan^{-1}(\frac{b}{a})
\]
Ensure to adjust \(\phi\) depending on the quadrant using signs of \(a\) and \(b\).
**Solution Input Boxes:**
- **Amplitude (\(A\)):**
Enter the calculated amplitude.
- **Phase Shift (\(\phi\)):**
Enter the calculated value of \(\phi\) (in radians).
Complete these calculations to find the values of \(A\) and \(\phi\). Make sure your calculator is set to the correct mode (degrees or radians) if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7513b29f-a9a0-42cc-81c6-8baa3bbf6943%2F68d5c3db-14fa-4244-a39e-9e7ae28e5cdc%2Fjqyadoh_processed.png&w=3840&q=75)
Transcribed Image Text:**Task: Convert a Trigonometric Expression**
Rewrite the expression \(-4 \sin(x) - 5 \cos(x)\) in the form \(A \sin(x + \phi)\).
**Solution Steps:**
1. **Amplitude Calculation (\(A\))**
Use the formula:
\[
A = \sqrt{a^2 + b^2}
\]
where \(a = -4\) and \(b = -5\).
2. **Phase Shift Calculation (\(\phi\))**
Use the arctangent to find \(\phi\):
\[
\phi = \tan^{-1}(\frac{b}{a})
\]
Ensure to adjust \(\phi\) depending on the quadrant using signs of \(a\) and \(b\).
**Solution Input Boxes:**
- **Amplitude (\(A\)):**
Enter the calculated amplitude.
- **Phase Shift (\(\phi\)):**
Enter the calculated value of \(\phi\) (in radians).
Complete these calculations to find the values of \(A\) and \(\phi\). Make sure your calculator is set to the correct mode (degrees or radians) if needed.
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