/*! 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
Question: Find all values of \(\theta\) in the interval \(0 \leq \theta < 2 \pi\) that satisfy the equation \(\sin \theta \cos \theta = 0\). Answer: The values of \(\theta\) that satisfy the equation in the given interval are \(0\), \(\pi\), \(\frac{\pi}{2}\), and \(\frac{3 \pi}{2}\).

Step by step solution

01

Set each factor equal to zero

We have the equation \(\sin \theta \cos \theta = 0\). According to the property mentioned above, one of the factors must be equal to zero. So we set up two separate equations: 1. \(\sin \theta = 0\) 2. \(\cos \theta = 0\) Now we must find all the values of \(\theta\) in the interval \(0 \leq \theta < 2 \pi\) that satisfy either Equation 1 or Equation 2.
02

Solve for \(\theta\) from Equation 1

From Equation 1 (\(\sin \theta = 0\)), we know that \(\theta\) must be the zero of the sine function within our given interval. The basic sine curve has zeroes at \(0\), \(\pi\), and \(2 \pi\). Since \(\theta\) must be within \(0 \leq \theta < 2 \pi\), the two values of \(\theta\) that satisfy this equation are \(\theta = 0\) and \(\theta = \pi\).
03

Solve for \(\theta\) from Equation 2

From Equation 2 (\(\cos \theta = 0\)), we need to find the zeroes of the cosine function within our given interval. The basic cosine curve has zeroes at leading values at \(\frac{\pi}{2}\) and \(\frac{3 \pi}{2}\). Therefore, we have two more values of \(\theta\) that satisfy this equation: \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3 \pi}{2}\).
04

Combine the solutions

We have found the possible values for \(\theta\) in our given interval: \(\theta = 0\), \(\theta = \pi\), \(\theta = \frac{\pi}{2}\), and \(\theta = \frac{3 \pi}{2}\). So, the solutions to the given equation, \(\sin \theta \cos \theta = 0\), in the interval \(0 \leq \theta < 2 \pi\) are \(\theta = 0\), \(\theta = \pi\), \(\theta = \frac{\pi}{2}\), and \(\theta = \frac{3 \pi}{2}\).

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