document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104061>Question 1104061</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find sin &#952; if cot &#952; = - 2 and cos &#952; < 0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #718801 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=718801\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> cot(T) = 1/tan(T), therefore:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1/tan(T) = -2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for tan(T) to get tan(T) = 1/-2 = -.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "tangent is negative in the second quadrant or the fourth quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cosine is negative in the second quadrant and third quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "therefore, if tangent is negative and cosine is negative, the angle has to be in the second quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the angle was in the first quadrant, the tangent would be positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "look for the angle in the first quadrant that has a tangent of .5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that angle would be equal to 26.56505118 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equivalent angle in the second quadrant would be 180 - that = 153.4349488 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the sine of that would be equal to .4472135953.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's your solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "another way:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "look at the triangle formed as if it was in the first quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the angle is in the first quadrant, than all trigonometric functions are positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that means that cotangent is equal to 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cotangent is equal to adjacent side / opposite side.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "therefore, adjacent side is equal to 2 and opposite side is equal to 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "by pythagorus, hypotenuse is equal to square root of (adjacent side squared plus opposite side squared).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this makes hypotenuse equal to sqrt(1^2 + 2^2) = sqrt(5).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sine is equal to opposite / hypotenuse which is equal to 1/sqrt(5).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the angle is in the second quadrant, then the sine is still positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "decimal equivalent of 1/sqrt(5) is equal to .4472135955.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "same answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "one more way of looking at it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you have determined that the angle has to be in the second quadrant, and you have determined that cotangent = -2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means that adjacent side (represented by x on the unit circle) / opposite side (represented by y in the unit circle) has to be equal to -2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "therefore x is equal to -2 and y is equal to 1 becuse -2/1 = -2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "draw your triangle in the second quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "find the measure of the hypotenuse, which is always positive, by using pythagorus and you get hypotenuse = sqrt(5).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for sine to get sine= 1/sqrt(5).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's the diagram of the unit circle with your triangle formed in the second quadranat.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/120801.jpg\" alt=\"$$$\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the adjacent side of the angle is the x-coordinate.<BR>\n" );
document.write( "the opposite side of the angle is the y-coordinate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cotangent theta = adjacent / opposite = -2/1 = -2, as required.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cosine theta = adjacent / hypotenuse = -2/sqrt(5) = negative as required.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sine theta = opposite / hypotenuse = 1/sqrt(5) = .4472135955 as we found above.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );