document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=889393>Question 889393</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My problem was (sin(&#960;/3) cos(&#960;/4)- sin(&#960;/4)cos(&#960;/3)² \r<BR>\n" );
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document.write( "I simplified that to (&#8730;6/4 - &#8730;2/4)² but I cant figure out what I should do next. Thank you for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #538158 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=538158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe this might be in the form of:\r<BR>\n" );
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document.write( "sin(a-b) = sin(a)cos(b) - cos(a) + sin(b), which is exactly the same as sin(a-b) = sin(a)cos(b) - sin(b)cos(a).\r<BR>\n" );
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document.write( "if so, then what you are really looking for is (sin(a-b)^2.\r<BR>\n" );
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document.write( "a-b would be equal to (pi/3 - pi/4) which would be equal to pi/12.\r<BR>\n" );
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document.write( "your problem then simplifies to (sin(pi/12))^2 which is equal to <BR>\n" );
document.write( ".0669872981.\r<BR>\n" );
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document.write( "we should also be able to work it through directly as you tried to do.\r<BR>\n" );
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document.write( "pi/3 = 60 degrees.<BR>\n" );
document.write( "pi/4 = 45 degrees.\r<BR>\n" );
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document.write( "sin(60) = sqrt(3)/2<BR>\n" );
document.write( "cos(60) = 1/2<BR>\n" );
document.write( "sin(45) = sqrt(2)/2<BR>\n" );
document.write( "cos(45) = sqrt(2)/2\r<BR>\n" );
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document.write( "the problem looks like the following in degrees.\r<BR>\n" );
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document.write( "(sin(60)cos(45) - sin(45)cos(60))^2\r<BR>\n" );
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document.write( "i re-arranged sin(45) and cos(60) but that shouldn't matter because a*b = b*a.\r<BR>\n" );
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document.write( "replace sin(60) with sqrt(3)/2 and replace cos(45) with sqrt(2)/2 and replace sin(45) with sqrt(2)/2 and replace cos(60) with 1/2 to get the following:\r<BR>\n" );
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document.write( "(sqrt(3)/2 * sqrt(2)/2 - sqrt(2)/2 * 1/2))^2\r<BR>\n" );
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document.write( "since sqrt(3) * sqrt(2) = sqrt(6), this simplifies to:\r<BR>\n" );
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document.write( "(sqrt(6)/4 - sqrt(2)/4)^2\r<BR>\n" );
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document.write( "this looks like where you were at.\r<BR>\n" );
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document.write( "it does not appear that you can combine these, so you need to perform the operation indicated.\r<BR>\n" );
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document.write( "that operation is to square the expression.\r<BR>\n" );
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document.write( "squaring that expression uses the distributive property of multiplication.\r<BR>\n" );
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document.write( "(sqrt(6)/4 - sqrt(2)/4)^2 is equivalent to (sqrt(6)/4 - sqrt(2)/4) * (sqrt(6)/4 - sqrt(2)/4)\r<BR>\n" );
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document.write( "now it's easier to see how to apply the distributive property.\r<BR>\n" );
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document.write( "the distributive propoerty is performed as follows:\r<BR>\n" );
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document.write( "(a-b)*(a-b) = a^2 - ab - ba + b^2 when then simplifies to a^2 - 2ab + b^2.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "6/16 - sqrt(6)/4 * sqrt(2)/4 - sqrt(2)/4 * sqrt(6)/4 + 2/16\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "6/16 - (sqrt(6)*sqrt(2))/16 - (sqrt(2)*sqrt(6))/16 + 2/16\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "8/16 - sqrt(12)/16 - sqrt(12)/16\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "(8 - sqrt(12) - sqrt(12)) / 16\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "(8 - 2*sqrt(12))/16\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "(4 - sqrt(12))/8\r<BR>\n" );
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document.write( "i believe you can simplify one step further to get:\r<BR>\n" );
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document.write( "1/2 - sqrt(12)/8\r<BR>\n" );
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document.write( "i don't believe you can simplify further than that.\r<BR>\n" );
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document.write( "your solution, in decimal form, would be .0669872981.\r<BR>\n" );
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document.write( "this is exactly the answer we got above when we used the sin(a-b) formula which confirms that the expression could have been converted up front, although doing that would have meant a decimal format answer without the corresponding exact answer because you would have been dealing with sin(15) degrees which doesn't translate easily to exact format, if at all.\r<BR>\n" );
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document.write( "in this case, it does translate because we know that .0669872981 came from the exact answer of 1/2 - sqrt(12)/8.   only problem is, if you started with .0669872981, how would you know that it came from 1/2 - sqrt(12)/8.  you wouldn't.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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