document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=921665>Question 921665</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Why is tan(-569) = -tan29 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #559180 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=559180\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> to understand why, you have to understand what reference angles are.\r<BR>\n" );
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document.write( "here's some references that might help you with that.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceAngles.htm\" target = \"_blank\">http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceAngles.htm</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceTriangles.htm\" target = \"_blank\">http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceTriangles.htm</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_4%20TRIF%20FNS%20OF%20ANGLES.pdf\" target = \"_blank\">http://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_4%20TRIF%20FNS%20OF%20ANGLES.pdf</a>\r<BR>\n" );
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document.write( "to make a long story short, the reference angle for -529 is equal to 29 degrees.\r<BR>\n" );
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document.write( "that is found as follows:\r<BR>\n" );
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document.write( "start with -569<BR>\n" );
document.write( "add 360 to get -209<BR>\n" );
document.write( "add 360 to get 151\r<BR>\n" );
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document.write( "an angle of 151 degrees occupies the same position in the unit circle that an angle of -569 degrees occupies.\r<BR>\n" );
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document.write( "they are effectively the same angles as far as the unit circle is concerned.\r<BR>\n" );
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document.write( "now that you are in the positive angle range of 0 to 360 degrees, you can find the reference angle.\r<BR>\n" );
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document.write( "the formula to find the reference angle is as follows:\r<BR>\n" );
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document.write( "if the angle is in quadrant 1, then the reference angle is the angle.<BR>\n" );
document.write( "if the angle is in quadrant 2, then the reference angle is 180 minus the angle.<BR>\n" );
document.write( "if the angle is in quadrant 3, then the reference angle is 180 plus the angle.<BR>\n" );
document.write( "if the angle is in quadrant 4, then the reference angle is 360 minus the angle.\r<BR>\n" );
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document.write( "151 is in quadrant 2, so the reference angle is 180 - 151 = 29 degrees.\r<BR>\n" );
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document.write( "the reference angle and the angle will have the same value for the trigonometric function with the exception of the sign.\r<BR>\n" );
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document.write( "in quadrant 1, all trigonometric functions are positive.<BR>\n" );
document.write( "in quadrant 2, sine is positive, cosine is negative, tangent is negative.<BR>\n" );
document.write( "in quadrant 3, sine is negative, cosine is negative, tangent is positive.<BR>\n" );
document.write( "in quadrant 4, sine is negative, cosine is positive, tangent is negative.\r<BR>\n" );
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document.write( "from the perspective of the tangent function, it is positive in quadrant 1 and negative in quadrant 2 and positive in quadrant 3 and negative in quadrant 4.\r<BR>\n" );
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document.write( "since the angle of 151 degrees is in quadrant 2, the tangent function is negative.\r<BR>\n" );
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document.write( "it has the same value as the reference angle.<BR>\n" );
document.write( "the sign, however, is different.<BR>\n" );
document.write( "this is why tangent of 151 is equal to minus the tangent of 29.\r<BR>\n" );
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document.write( "if you understand the  unit circle, you will understand why the signs change.\r<BR>\n" );
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document.write( "let h represent hypotenuse<BR>\n" );
document.write( "in the unit circle, y represents the side opposite the angle and x represents the side adjacent the angle.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "sine = y/h<BR>\n" );
document.write( "cosine = x/h<BR>\n" );
document.write( "tangent = y/x\r<BR>\n" );
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document.write( "in quadrant 1, x and y are positive.<BR>\n" );
document.write( "in quadrant 2, x is negative and y is positive.<BR>\n" );
document.write( "in quadrant 3, x and y are negative.<BR>\n" );
document.write( "in quadrant 4, x is positive and y is negative.\r<BR>\n" );
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document.write( "this is why the trig functions have different signs, depending on the quadrant they are in.\r<BR>\n" );
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document.write( "in quadrant 1, tangent = y/x = +/+ = +<BR>\n" );
document.write( "in quadrant 2, tangent = y/x = +/- = -\r<BR>\n" );
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document.write( "so, you have tan(151) = -tan(29).\r<BR>\n" );
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document.write( "the following graph of the tangent function will show you that the angles have the same tangent value every 180 degrees.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/110903.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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