document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198723>Question 1198723</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given cot theta =-2 and csc theta < 0, find sin theta and sec theta </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832552 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=832552\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "In response to what the tutor @MathLover1 wrote: Cotangent is NOT negative in quadrant Q3.<BR>\n" );
document.write( "Take note how sine and cosine are indeed negative here, so,<BR>\n" );
document.write( "cot = cos/sin<BR>\n" );
document.write( "the two negative items divide to a positive result.\r<BR>\n" );
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document.write( "In short, cotangent is positive in Q3.\r<BR>\n" );
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document.write( "Since cot(theta) = -2 is negative, this places theta in either Q2 or Q4.\r<BR>\n" );
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document.write( "Then we're told csc(theta) < 0, which fully narrows things down to Q4 only.<BR>\n" );
document.write( "270 < theta < 360 in degree mode<BR>\n" );
document.write( "3pi/2 < theta < 2pi in radian mode<BR>\n" );
document.write( "The actual theta value itself doesn't matter; however, this interval helps narrow things down a bit.\r<BR>\n" );
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document.write( "Since cot(theta) = -2 = 2/(-1), we could have a triangle with adjacent side 2 and opposite side -1.<BR>\n" );
document.write( "I'll make the opposite length \"negative\" so that we can keep the proper signs in mind. Of course a negative length isn't possible. <BR>\n" );
document.write( "It's purely as a means to retain information.<BR>\n" );
document.write( "I'm making the 1 negative so that it indicates we're below the x axis, i.e. the y coordinate is negative.\r<BR>\n" );
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document.write( "Use the pythagorean theorem to find the hypotenuse is sqrt(5) units long.\r<BR>\n" );
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document.write( "This is one way to draw the triangle in Q4<BR>\n" );
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document.write( "We have:<BR>\n" );
document.write( "opposite = -1<BR>\n" );
document.write( "adjacent = 2<BR>\n" );
document.write( "hypotenuse = sqrt(5)\r<BR>\n" );
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document.write( "Then recall that<BR>\n" );
document.write( "sin(theta) = opposite/hypotenuse<BR>\n" );
document.write( "sec(theta) = hypotenuse/adjacent\r<BR>\n" );
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document.write( "I'll let you finish up from here.<BR>\n" );
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