document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=667427>Question 667427</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find sin 2x, cos 2x, and tan 2x from the given information.<BR>\n" );
document.write( "csc x = 4,   tan x < 0<BR>\n" );
document.write( "sin 2x	 =	 \r<BR>\n" );
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document.write( "cos 2x	 =	<BR>\n" );
document.write( "tan 2x	 =	 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #415033 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=415033\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \csc\left(\varphi\right)\ =\ \frac{1}{\sin\left(\varphi\right)}\">\r<BR>\n" );
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document.write( "Hence, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sin(x)\ =\ \frac{1}{4}\">\r<BR>\n" );
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document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos\left(\varphi\right)^2\ +\sin\left(\varphi\right)^2\ =\ 1\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(x)\ =\ \pm\frac{\sqrt{15}}{4}\">.\r<BR>\n" );
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document.write( "But since the tangent is the sine divided by the cosine and we have a positive sine value, and are given that the tangent is negative, the cosine must be negative. Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos(x)\ =\ -\frac{\sqrt{15}}{4}\">\r<BR>\n" );
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document.write( "Using the double angle formulae:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin\left(2\varphi\right)\ =\ 2\sin\left(\varphi\right)\cos\left(\varphi\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin\left(2x\right)\ =\ 2\left(\frac{1}{4}\right)\left(-\frac{\sqrt{15}}{4}\right)\ =\ -\frac{\sqrt{15}}{8}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\left(2\varphi\right)\ =\ \cos^2\left(\varphi\right)\ -\ \sin^2\left(\varphi\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos\left(2x\right)\ =\ \frac{15}{16}\ -\ \frac{1}{16}\ =\ \frac{7}{8}\">\r<BR>\n" );
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document.write( "Then since:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \tan\left(\varphi\right)\ =\ \frac{\sin\left(\varphi\right)}{\cos\left(\varphi\right)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \tan\left(2x\right)\ =\ \frac{\sin\left(2x\right)}{\cos\left(2x\right)}\ =\ \frac{\frac{-\sqrt{15}}{8}}{\frac{7}{8}}\ =\ -\frac{\sqrt{15}}{7}\">\r<BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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