document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197407>Question 1197407</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If θ is an angle in the third quadrant and csc θ=-(13/5), find the value of 3sin^2 θ - 2cos θ as a fraction with no decimals. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830667 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=830667\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If you use the pythagorean theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you'll have a 5-12-13 right triangle.<BR>\n" );
document.write( "leg1 = 5 units<BR>\n" );
document.write( "leg2 = 12 units<BR>\n" );
document.write( "hypotenuse = 13 units\r<BR>\n" );
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document.write( "Start at the origin of the xy coordinate axis.<BR>\n" );
document.write( "Move 12 units to the left, then move 5 units down<BR>\n" );
document.write( "This forms a right triangle with the 90 degree angle on the x axis.<BR>\n" );
document.write( "Angle theta is shown in green below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-15%2C15%2C-9%2C9%2C%0D%0Agraph%28500%2C500%2C-15%2C15%2C-9%2C9%2C0%29%2C%0D%0Aline%280%2C0%2C-12%2C0%29%2C%0D%0Aline%28-12%2C0%2C-12%2C-5%29%2C%0D%0Aline%28-12%2C-5%2C0%2C0%29%2C%0D%0Alocate%28-6%2C1%2C%22-12%22%29%2C%0D%0Alocate%28-13.5%2C-2.5%2C%22-5%22%29%2C%0D%0Agreen%28arc%28+0%2C+0%2C+5%2B0%2C+4%2B0%2C+360-202.61986494804%2C0+%29%29%2C%0D%0Agreen%28arc%28+0%2C+0%2C+5%2B0.1%2C+4%2B0.1%2C+360-202.61986494804%2C0+%29%29%2C%0D%0Agreen%28arc%28+0%2C+0%2C+5%2B0.2%2C+4%2B0.2%2C+360-202.61986494804%2C0+%29%29%2C%0D%0Agreen%28arc%28+0%2C+0%2C+5%2B0.3%2C+4%2B0.3%2C+360-202.61986494804%2C0+%29%29%2C%0D%0Agreen%28arc%28+0%2C+0%2C+5%2B0.4%2C+4%2B0.4%2C+360-202.61986494804%2C0+%29%29%2C%0D%0Agreen%28arc%28+0%2C+0%2C+5%2B0.5%2C+4%2B0.5%2C+360-202.61986494804%2C0+%29%29%2C%0D%0Alocate%282.5%2C2%2Cgreen%28theta%29%29%2C%0D%0Alocate%28-7%2C-3%2C%2213%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We start facing directly east along the positive x axis.<BR>\n" );
document.write( "Then rotate until we are aiming in the southwest direction along the hypotenuse of length 13.\r<BR>\n" );
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document.write( "Csc = cosecant<BR>\n" );
document.write( "csc is the reciprocal of sine\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=csc%28theta%29+=+-13%2F5 ALIGN=MIDDLE  ALT=\"csc%28theta%29+=+-13%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28theta%29+=+-5%2F13 ALIGN=MIDDLE  ALT=\"sin%28theta%29+=+-5%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then use the pythagorean trig identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%28theta%29%2Bcos%5E2%28theta%29+=+1 ALIGN=MIDDLE  ALT=\"sin%5E2%28theta%29%2Bcos%5E2%28theta%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > with that sine value found to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29+=+-12%2F13 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+-12%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Or you can use the reference diagram above.\r<BR>\n" );
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document.write( "sine = opposite/hypotenuse<BR>\n" );
document.write( "cosine = adjacent/hypotenuse<BR>\n" );
document.write( "In this case, the opposite side is the vertical leg labeled 5. The adjacent side is 12 units long. <BR>\n" );
document.write( "The negative signs are there so we get the correct sign for each trig ratio.\r<BR>\n" );
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document.write( "Now we can compute what your teacher asked<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asin%5E2%28theta%29+-+2%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"3%2Asin%5E2%28theta%29+-+2%2Acos%28theta%29\"   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=3%2A%28-5%2F13%29%5E2+-+2%2A%28-12%2F13%29 ALIGN=MIDDLE  ALT=\"3%2A%28-5%2F13%29%5E2+-+2%2A%28-12%2F13%29\"   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=3%2A%2825%2F169%29+-+2%2A%28-12%2F13%29 ALIGN=MIDDLE  ALT=\"3%2A%2825%2F169%29+-+2%2A%28-12%2F13%29\"   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=75%2F169+%2B24%2F13 ALIGN=MIDDLE  ALT=\"75%2F169+%2B24%2F13\"   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=75%2F169+%2B%2824%2A13%29%2F%2813%2A13%29 ALIGN=MIDDLE  ALT=\"75%2F169+%2B%2824%2A13%29%2F%2813%2A13%29\"   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=%2875+%2B312%29%2F169 ALIGN=MIDDLE  ALT=\"%2875+%2B312%29%2F169\"   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=387%2F169 ALIGN=MIDDLE  ALT=\"387%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer: <font color=red>387/169</font><BR>\n" );
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