/*! 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 11 Solve the equation. $$\sqrt{3}... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 11

Solve the equation. $$\sqrt{3} \csc x-2=0$$

Short Answer

Expert verified
The solution to the equation is \(x = \pi/3 + n2\pi\) and \(x = 2\pi/3 + n2\pi\), where \(n\) is an integer.

Step by step solution

01

Simplify the equation

First, simplify \(\sqrt{3} \csc x-2=0\) to \(\csc x = 2/\sqrt{3}\). You can do this by adding 2 to both sides and then dividing by \(\sqrt{3}\). Now, since \(\csc x\) is the reciprocal of \(\sin x\), we can rewrite the equation as \(\sin x = \sqrt{3}/2\).
02

Use the properties of sin(x)

We know that \(\sin x = \sqrt{3}/2\) at \(x = \pi/3\) and \(x = 2\pi/3\) in the interval of [0,2\pi]. Since the sin function has a period of 2\pi, solutions for \(x\) will occur every 2\pi. Hence, the general solutions of the equation can be given as \(x = \pi/3 + n2\pi\) and \(x = 2\pi/3 + n2\pi\), where \(n\) is an integer.
03

Write down the final solution

After finding the general solutions, write down the answer as \(x = \pi/3 + n2\pi\) and \(x = 2\pi/3 + n2\pi\), \(n\) integer.

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