document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=754320>Question 754320</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let theta be an angle in quadrant 3 such that the csc of theta = -6/5.<BR>\n" );
document.write( "Find the exact values of the tan of theta and the cos of theta.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #458985 by </B><A HREF=/tutors/aboutme.mpl?userid=anteater><B>anteater(7)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=anteater><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=458985\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can start by drawing a right triangle in the third quadrant. They tell you that the cosecant of theta is -6/5. The cosecant is the ratio of the length of the hypotenuse to the length of the side opposite theta.\r<BR>\n" );
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document.write( "The hypotenuse always has a positive value, and you would label the hypotenuse \"6\" and the side opposite the angle \"-5\". You will then also want to find the adjacent side of your triangle (the side lying on the x-axis). Use the Pythagorean theorem to find this side.\r<BR>\n" );
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document.write( "You know a^2 + b^2 = c^2. For your triangle, then, (-5)^2 + b^2 = 6^2\r<BR>\n" );
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document.write( "25 + b^2 = 36 --> b^2 = 11 --> b = sqrt(11). And, since your triangle is in the 3rd quadrant, the adjacent side of your triangle will have a negative sign, so b = -sqrt(11)\r<BR>\n" );
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document.write( "So you have:\r<BR>\n" );
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document.write( "hypotenuse = 6\r<BR>\n" );
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document.write( "opposite = -5\r<BR>\n" );
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document.write( "adjacent = -sqrt(11)\r<BR>\n" );
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document.write( "You can use these values to find the tangent of theta and the cosine of theta.\r<BR>\n" );
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document.write( "Tan (theta) = opposite/adjacent = -5/-sqrt(11)\r<BR>\n" );
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document.write( "Since this has a radical expression in the denominator, they may want you to \"rationalize\" the denominator. To do that, you could multiply by sqrt(11)/sqrt(11):\r<BR>\n" );
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document.write( "(-5/-sqrt(11)) X (sqrt(11)/sqrt(11)) = 5sqrt(11)/11     This should be the tangent of your angle.\r<BR>\n" );
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document.write( "Cos (theta) = adjacent/hypotenuse  = -sqrt(11)/6    This should be the cosine of your angle.\r<BR>\n" );
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document.write( "I hope this was helpful!\r<BR>\n" );
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