/*! 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 66 find the solutions of the equati... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 66

find the solutions of the equation in \([0,2 \pi)\). $$ (\sin 2 x)(\sin x)=0 $$

Short Answer

Expert verified
The solutions of the equation \((\sin(2x))(\sin(x))=0\) in the interval \([0, 2\pi)\) are \(x=0\), \(x=\frac{\pi}{2}\), \(x=\pi\), and \(x=\frac{3\pi}{2}\).

Step by step solution

01

Set each factor equal to zero

We have the equation: \((\sin(2x))(\sin(x))=0\) In order to solve this equation, we can set each factor equal to zero: \(\sin(2x)=0\) or \(\sin(x)=0\)
02

Solve for \(x\) in \(\sin(2x)=0\)

For the first part of the solution, we solve the equation \(\sin(2x) = 0\) for \(x\) in the interval of \([0, 2\pi)\). \(\sin(2x)=0\) In order to solve this equation, we need to find all possible values of \(x\) for which \(\sin(2x)=0\) in the given interval. The sine function is equal to zero at all integral multiples of \(\pi\), so in this case: \(2x=n\pi\), where \(n\) is an integer. Divide by 2 to isolate \(x\): \(x=\frac{n\pi}{2}\) Note that in this case, \(x\) will be within the interval \([0, 2\pi)\) for integer values of \(n\) such that \(0\leq n \leq 3\). Thus, the solutions for \(x\) in this case are: \(x=0\), \(x=\frac{\pi}{2}\), \(x=\pi\), and \(x=\frac{3\pi}{2}\).
03

Solve for \(x\) in \(\sin(x)=0\)

For the second part of the solution, we solve the equation \(\sin(x) = 0\) for \(x\) in the interval of \([0, 2\pi)\). \(\sin(x)=0\) In this case, we can directly find the values of \(x\) for which \(\sin(x)=0\) in the given interval since the sine function is equal to zero at all integral multiples of \(\pi\): \(x=n\pi\), where \(n\) is an integer. Within the given interval, only \(n=0\) and \(n=1\) will give valid values for \(x\). Hence, the solutions for \(x\) in this case are: \(x=0\) and \(x=\pi\).
04

Combine the solutions

Now we have found all the possible solutions for \(x\) from both equations in the given interval \([0, 2\pi)\). We can combine these solutions to get the complete set of solutions for the original equation: \(x=0\), \(x=\frac{\pi}{2}\), \(x=\pi\), and \(x=\frac{3\pi}{2}\).

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