document.write( "Question 788739: find all solutions in the interval [0,2pi) for the equation 3csc x-4sin x=0. \n" ); document.write( "

Algebra.Com's Answer #478595 by KMST(5328) $\"\"$ $\"About$

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$\"csc%28x%29=1%2Fsin%28x%29\"$ does not exist when $\"sin%28x%29=0\"$ at $\"x=0\"$ and $\"x=pi\"$
\n" ); document.write( "For any other values of $\"x\"$ , we can transform the equation into an equivalent quadratic equation by multiplying times $\"sin%28x%29\"$ :
\n" ); document.write( " $\"3csc%28x%29-4sin%28x%29=0\"$ <--> $\"3%2Fsin%28x%29-4sin%28x%29=0\"$ --> $\"3-4%28sin%28x%29%29%5E2=0\"$
\n" ); document.write( " $\"3-4%28sin%28x%29%29%5E2=0\"$ --> $\"3=4%28sin%28x%29%29%5E2=0\"$ --> $\"%28sin%28x%29%29%5E2=0=+3%2F4\"$
\n" ); document.write( "The solutions come from
\n" ); document.write( " $\"sin%28x%29=sqrt%283%2F4%29=sqrt%283%29%2F2\"$ and
\n" ); document.write( " $\"sin%28x%29=-sqrt%283%2F4%29=-sqrt%283%29%2F2\"$
\n" ); document.write( "In the interval $\"%22%5B0%2C%22\"$ $\"2pi\"$ $\"%22%29%22\"$
\n" ); document.write( " $\"sin%28x%29=sqrt%283%29%2F2\"$ --> $\"system%28highlight%28x=pi%2F3%29%2C+%22or%22%2Chighlight%28x=2pi%2F3%29%29\"$ ( $\"sin%28pi%2F3%29=sin%2860%5Eo%29=sqrt%283%29%2F2\"$ and being a supplementary angle $\"2pi%2F3=pi-pi%2F3\"$ has the same sine)
\n" ); document.write( "and $\"sin%28x%29=-sqrt%283%29%2F2\"$ --> $\"system%28highlight%28x=4pi%2F6%29%2C+%22or%22%2Chighlight%28x=5pi%2F6%29%29\"$ (Those are co-terminal angles of $\"-pi%2F3\"$ and $\"-2pi%2F3\"$ , found adding $\"2pi\"$ to $\"-pi%2F3\"$ and $\"-2pi%2F3\"$ )
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\n" ); document.write( "NOTE:
\n" ); document.write( "For this kind of equation, a change of variable often helps to see the equation as a quadratic equation.
\n" ); document.write( "It is easier to see $\"3-4%28sin%28x%29%29%5E2=0\"$ as a quadratic equation using the change of variable $\"sin%28x%29=y\"$ .
\n" ); document.write( "In this case, it was easy to solve the problem, even without mentioning a quadratic equation. In other cases, solving the quadratic may be more complivcated, so making the change of variable may save confusion and ink.
\n" ); document.write( "We get $\"3-4y%5E2=0\"$ \n" ); document.write( "

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