/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 45

Solve the following equations. $$\sin \theta \cos \theta=0,0 \leq \theta<2 \pi$$

Short Answer

Expert verified
Answer: The values of \(\theta\) that satisfy the equation are \(0, \pi/2, \pi, \text{ and } 3\pi/2\).

Step by step solution

01

Set up the possible cases

Since \(\sin \theta \cdot \cos \theta = 0\), either \(\sin \theta = 0\) or \(\cos \theta = 0\) or both. So we need to find the values of \(\theta\) that satisfy each case in the given range.
02

Solve for case when sin 饾渻 = 0

Recall that \(\sin \theta = 0\) when \(\theta = n\pi\), where \(n\) is an integer. So, let's find the integer values of \(n\) in the range of \(0 \leq \theta < 2 \pi\). When \(n = 0\), we have \(\theta = 0\), and when \(n = 1\), we get \(\theta = \pi\). When \(n = 2\), we get \(\theta = 2\pi\), which is outside our range. So the solutions for this case are \(\theta = 0\) and \(\theta = \pi\).
03

Solve for case when cos 饾渻 = 0

Recall that \(\cos \theta = 0\) when \(\theta = \pi / 2 + n\pi\), where \(n\) is an integer. So let's find the integer values of \(n\) in the range \(0 \leq \theta < 2 \pi\). When \(n = 0\), we have \(\theta = \pi / 2\), and when \(n = 1\), we get \(\theta = 3\pi / 2\). When \(n = 2\), we get \(\theta = 5\pi / 2\), which is outside our range. So the solutions for this case are \(\theta = \pi / 2\) and \(\theta = 3\pi / 2\).
04

Combine the solutions

We have four solutions for \(\theta\) that satisfy either \(\sin \theta = 0\) or \(\cos \theta = 0\). Combining the solutions from both cases gives us: $$\theta = 0, \pi/2, \pi, \text{ and } 3\pi/2$$

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