document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196692>Question 1196692</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what happens to sinθ as θ approaches 0 radians? Why? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829633 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20855)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way most people first encounter sine is as a ratio of two sides of a triangle.\r<BR>\n" );
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document.write( "In this case:\r<BR>\n" );
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document.write( "1. the sloping lines are all radii of the circle and represent the hypotenuse<BR>\n" );
document.write( "2. the vertical lines are the ‘opposite’ side of the various triangles\r<BR>\n" );
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document.write( "As you can clearly see, as the angle reduces from \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=85 ALIGN=MIDDLE  ALT=\"85\"   DISABLED_style= \"max-width:90%; height:auto;\" >°  → <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >° → <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45 ALIGN=MIDDLE  ALT=\"45\"   DISABLED_style= \"max-width:90%; height:auto;\" >° → <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30 ALIGN=MIDDLE  ALT=\"30\"   DISABLED_style= \"max-width:90%; height:auto;\" >° →<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" >° →<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" >° →<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >° →<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >°  \r<BR>\n" );
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document.write( "the height of the vertical line (’opposite’) shrinks while the sloping line (‘hypotenuse’) remains constant \r<BR>\n" );
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document.write( " So the fraction  opposite hypotenuse will, does and indeed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=must+ ALIGN=MIDDLE  ALT=\"must+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=reduce ALIGN=MIDDLE  ALT=\"reduce\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > as  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >  reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "The more modern way to define sine is with an infinite series:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29=x-x%5E3%2F3%21%2Bx%5E5%2F5%21-x%2A7%2F7%21 ALIGN=MIDDLE  ALT=\"sin%28x%29=x-x%5E3%2F3%21%2Bx%5E5%2F5%21-x%2A7%2F7%21\"   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=sin%28x%29=x%281-x%5E2%2F3%21%2Bx%5E4%2F5%21-x%5E6%2F7%21%2Bx%5E8%2F9%21%29 ALIGN=MIDDLE  ALT=\"sin%28x%29=x%281-x%5E2%2F3%21%2Bx%5E4%2F5%21-x%5E6%2F7%21%2Bx%5E8%2F9%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.... \r<BR>\n" );
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document.write( "But you can see how that also goes to zero as  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >  goes to zero!\r<BR>\n" );
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