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 1: Find all solutions to \( \cos(3\pi x) = \frac{1}{2} \).**
To solve this equation, we look for the angles where the cosine function equals \(\frac{1}{2}\). The general solution for \(\cos \theta = \frac{1}{2}\) occurs at angles:
\[
\theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \theta = -\frac{\pi}{3} + 2k\pi,
\]
where \(k\) is an integer.
Substituting \(3\pi x\) in place of \(\theta\), we have:
1. \(3\pi x = \frac{\pi}{3} + 2k\pi\)
2. \(3\pi x = -\frac{\pi}{3} + 2k\pi\)
Solving for \(x\):
1. \(x = \frac{1}{9} + \frac{2k}{3}\)
2. \(x = -\frac{1}{9} + \frac{2k}{3}\)
where \(k\) is any integer. These equations give the general solutions for the original problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6735299-61de-45a1-8389-cd026716d460%2F37ab182f-9c38-4ff1-b45c-7316adfd5064%2Fynmezyv_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1: Find all solutions to \( \cos(3\pi x) = \frac{1}{2} \).**
To solve this equation, we look for the angles where the cosine function equals \(\frac{1}{2}\). The general solution for \(\cos \theta = \frac{1}{2}\) occurs at angles:
\[
\theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \theta = -\frac{\pi}{3} + 2k\pi,
\]
where \(k\) is an integer.
Substituting \(3\pi x\) in place of \(\theta\), we have:
1. \(3\pi x = \frac{\pi}{3} + 2k\pi\)
2. \(3\pi x = -\frac{\pi}{3} + 2k\pi\)
Solving for \(x\):
1. \(x = \frac{1}{9} + \frac{2k}{3}\)
2. \(x = -\frac{1}{9} + \frac{2k}{3}\)
where \(k\) is any integer. These equations give the general solutions for the original problem.
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