document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=281690>Question 281690</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all points [r, theta] of intersection r=1 and r= 2sin(theta)\r<BR>\n" );
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document.write( "Multiple choice:<BR>\n" );
document.write( "a)[1/2, pi/3] & [1/2, 5pi/3]<BR>\n" );
document.write( "b) [1/2, pi/6] and [1/2, 5pi/6]<BR>\n" );
document.write( "c) [1, pi/3] & [1,5pi/3]<BR>\n" );
document.write( "d) [1, pi/6] and [1,5pi/6]<BR>\n" );
document.write( "e) None of these\r<BR>\n" );
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document.write( "I want to eliminate A & B because their r-values are 1/2 instead of 1. \r<BR>\n" );
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document.write( "If I graph these in Polar mode (in radian mode too), the two graphs form two ellipses.  If I trace, I can find intersections at theta = .5236 (pi/6) and 2.618 (5pi/6).  So I'm leading toward D as my answer. \r<BR>\n" );
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document.write( "First, is it okay to use the graphical approach to solving this problem?<BR>\n" );
document.write( "Second, what is the algebraic approach to solving this problem? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #204655 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=204655\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There's nothing wrong with using a graphical approach. You just need to keep in mind that the solutions will be approximate, which means that you need to figure out a way to get them into exact form. Since you are familiar with 0.5236 and 2.618 as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F6 ALIGN=MIDDLE  ALT=\"pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5pi%2F6 ALIGN=MIDDLE  ALT=\"5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the second concern isn't really an issue. However, the approximate solutions may not be so easy to recognize in general. Because of this potential problem, I suggest to use an algebraic approach.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=2sin%28theta%29 ALIGN=MIDDLE  ALT=\"r=2sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=2sin%28theta%29 ALIGN=MIDDLE  ALT=\"1=2sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1 ALIGN=MIDDLE  ALT=\"r=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2=sin%28theta%29 ALIGN=MIDDLE  ALT=\"1%2F2=sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%28theta%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F6 ALIGN=MIDDLE  ALT=\"theta=pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=5pi%2F6 ALIGN=MIDDLE  ALT=\"theta=5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the arcsine of both sides\r<BR>\n" );
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document.write( "Note: Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F6%29=sin%285pi%2F6%29=1%2F2 ALIGN=MIDDLE  ALT=\"sin%28pi%2F6%29=sin%285pi%2F6%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the arcsine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F6 ALIGN=MIDDLE  ALT=\"pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5pi%2F6 ALIGN=MIDDLE  ALT=\"5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So the exact solutions for theta are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=pi%2F6 ALIGN=MIDDLE  ALT=\"theta=pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=5pi%2F6 ALIGN=MIDDLE  ALT=\"theta=5pi%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So you are correct. </SPAN>\n" );
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