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 Statement**
Solve the equation:
\[ 4 \cos(2\phi) = 4 \sin^2(\phi) + 3 \]
for all solutions where \( 0 \leq \phi < 2\pi \).
**Solution Box**
\[ \phi = \]
**Explanation**
This problem is asking to solve a trigonometric equation within the interval from 0 to \(2\pi\). The left side of the equation involves the double angle cosine function, \(\cos(2\phi)\), while the right side features a squared sine term.
To solve this, consider trigonometric identities such as the double-angle identities:
\[ \cos(2\phi) = 1 - 2\sin^2(\phi) \text{ or } \cos(2\phi) = 2\cos^2(\phi) - 1 \]
By using these identities, you can transform the equation into a quadratic form or another solvable form to find all possible values of \(\phi\) that satisfy the equation within the given range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6735299-61de-45a1-8389-cd026716d460%2Ff27e8d85-7543-4eed-988d-978254ce8d96%2Fbqld0qp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Solve the equation:
\[ 4 \cos(2\phi) = 4 \sin^2(\phi) + 3 \]
for all solutions where \( 0 \leq \phi < 2\pi \).
**Solution Box**
\[ \phi = \]
**Explanation**
This problem is asking to solve a trigonometric equation within the interval from 0 to \(2\pi\). The left side of the equation involves the double angle cosine function, \(\cos(2\phi)\), while the right side features a squared sine term.
To solve this, consider trigonometric identities such as the double-angle identities:
\[ \cos(2\phi) = 1 - 2\sin^2(\phi) \text{ or } \cos(2\phi) = 2\cos^2(\phi) - 1 \]
By using these identities, you can transform the equation into a quadratic form or another solvable form to find all possible values of \(\phi\) that satisfy the equation within the given range.
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