document.write( "Question 1116853: Hi tutors, this is a extra points question for a quiz, but my professor never teaches us this kind of question.\r
\n" ); document.write( "\n" ); document.write( "Find the exact general and particular solutions to the equation:
\n" ); document.write( "10sin ( 3x+ pi/6 ) + 7sqrt(3) = 2sqrt(3)
\n" ); document.write( "1. What is the exact general solution?
\n" ); document.write( "2. What is the particular solutions in {0, 2pi }\r
\n" ); document.write( "\n" ); document.write( "can someone help me with this question ?
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Algebra.Com's Answer #731774 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
1. An exact general solution \"highlight%28x=%2812n%2B8+%2B-+1%29pi%2F18%29\" .
\n" ); document.write( "2. The particular solutions in \"%22%5B+0+%2C%22\"\"2pi\"\"%22%29%22\"
\n" ); document.write( "can found by using the formula above for \"%22n+=+0+%2C+1+%2C+2%22\"
\n" ); document.write( "\"x=highlight%287pi%2F18%29\" and \"x=9pi%2F18=highlight%28pi%2F2%29\" for \"n=0\" ,
\n" ); document.write( "\"x=highlight%2819pi%2F18%29\" and \"x=21pi%2F18=highlight%287pi%2F6%29\" for \"n=1\" , and
\n" ); document.write( "\"x=highlight%2831pi%2F18%29\" and \"x=33pi%2F18=highlight%2811pi%2F6%29\" for \"n=2\" .\r
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\n" ); document.write( "\n" ); document.write( "EXPLANATION AND GRAPHICAL REORESENTATION:
\n" ); document.write( "As Edwin showed you, simple algebra leads you to
\n" ); document.write( "\"sin%283x%2B+pi%2F6%29=-sqrt%283%29%2F2\" .
\n" ); document.write( "
\n" ); document.write( "What angles could measure \"3x%2B+pi%2F6\" ?
\n" ); document.write( "
\n" ); document.write( "The angles \"AOB\" and \"AOD\" have
\n" ); document.write( "\"sin%28AOB%29=sin%28AOD%29=-sqrt%283%29%2F2=about-0.866\" .
\n" ); document.write( "One whole counterclockwise turn is \"2pi\" (radians),
\n" ); document.write( "\"AOC=3pi%2F2\" .
\n" ); document.write( "Angles \"AOB\" and \"AOD\" measure \"pi%2F6\" less, and \"pi%2F6\" more than that,
\n" ); document.write( "or \"4pi%2F3\" and \"5pi%2F3\" respectively
\n" ); document.write( "A general formula for clockwise expression for the measure of \"AOB\" or \"AOD\" is
\n" ); document.write( "\"3pi%2F2+%2B-+pi%2F6=%289+%2B-+1%29pi%2F6\" ,
\n" ); document.write( "and a general formula for all angles co-terminal with those is
\n" ); document.write( "\"%289+%2B-+1%29pi%2F6+%2B2n%2Api=%2812n%2B9+%2B-+1%29pi%2F6\" for any integer \"n\" .
\n" ); document.write( "So, to find the exact general solution, with \"n\" representing any integer
\n" ); document.write( "\"3x%2B+pi%2F6=%2812n%2B9+%2B-+1%29pi%2F6\"
\n" ); document.write( "\"3x=%2812n%2B8+%2B-+1%29pi%2F6+-+pi%2F6\"
\n" ); document.write( "\"3x=%2812n%2B8+%2B-+1%29pi%2F6\"
\n" ); document.write( "\"x=%2812n%2B8+%2B-+1%29pi%2F18\"\r
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\n" ); document.write( "\n" ); document.write( "RANT:
\n" ); document.write( "In math, there is no such a thing as \"this kind of question\"
\n" ); document.write( "or \"this kind of problem\".
\n" ); document.write( "Unfortunately, too many math instructors and tutors
\n" ); document.write( "classify problems into hundreds or thousands of \"kinds\" or \"types\",
\n" ); document.write( "and then encourage students to memorize the classification,
\n" ); document.write( "and the corresponding problem-solving recipes.
\n" ); document.write( "That makes math into a no-reasoning memorization exercise similar to learning a foreign language.
\n" ); document.write( "Why?
\n" ); document.write( "Maybe because is easier than trying to persuade students that they only need to
\n" ); document.write( "1) understand a few simple concepts, and
\n" ); document.write( "2) apply those concepts and their own brains.
\n" ); document.write( "Maybe because they were taught math that same way,
\n" ); document.write( "and they do not know any better.
\n" ); document.write( "Or is it that they believe the students are incapable of reasoning?
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