/*! 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 35 Solve the equation. $$\sin x(\... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 35

Solve the equation. $$\sin x(\sin x+1)=0$$

Short Answer

Expert verified
The solutions of the equation \(\sin x(\sin x+1)=0\) are \(x = n\pi\) and \(x = (2n+1)\pi + 3\pi/2\), for all integers n.

Step by step solution

01

Set each factor equal to zero

The equation \(\sin x(\sin x+1)=0\) can be split into two separate equations, using the principle that a product of factors is zero if and only if at least one of the factors is zero. Thus, the two equations will be \(\sin x = 0\) and \(\sin x + 1 = 0\).
02

Solve for x in the first equation

In our first equation, \(\sin x = 0\), the values of x that satisfy this condition are \(x = n\pi\), where n is an integer. This is because the sine function oscillates between -1 and 1 and is equal to zero at every multiple of \(\pi\).
03

Solve for x in the second equation

For the second equation, check the values of x where the sine function equals -1. Subtract 1 from both sides of the equation \(\sin x + 1 = 0\) to get \(\sin x = -1\). The sine function equals -1 at \(x = (2n+1)\pi + 3\pi/2\), where n is any integer.
04

Combine all solutions

After solving both equations, combine all the values of x from steps 2 and 3 to obtain the complete solution set. The solutions are \(x = n\pi\) and \(x = (2n+1)\pi + 3\pi/2\), for all integers n.

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