document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084511>Question 1084511</A>: <I><SPAN STYLE=\"word-break: break-all;\"> True or false: sin (x-180) = sin (180-x)\r<BR>\n" );
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document.write( "Could you explain this using the unit circle?\r<BR>\n" );
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document.write( "Sincerely,<BR>\n" );
document.write( "Just a student trying to learn some maths </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698607 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=698607\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sine (180 - x) would give you the angle in the second quadrant, assuming the angle is originally in the first quadrant.\r<BR>\n" );
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document.write( "in other words, x is the angle in the first quadrant, therefore 180 - x is the equivalent angle in the second quadrant.\r<BR>\n" );
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document.write( "if x is in the first quadrant, then x - 180 would be the equivalent angle in the third quadrant.\r<BR>\n" );
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document.write( "sin(x) would be positive.\r<BR>\n" );
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document.write( "sin(180-x) would be also positive.\r<BR>\n" );
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document.write( "sin(x-180) would be negative.\r<BR>\n" );
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document.write( "this is because sine is positive in first and second quadrant and negative in second and third quadrant.\r<BR>\n" );
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document.write( "let's take an example:.\r<BR>\n" );
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document.write( "assume x = 30 degrees.\r<BR>\n" );
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document.write( "sin(30) = 1/2\r<BR>\n" );
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document.write( "180 - 30 would be equal to 150.\r<BR>\n" );
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document.write( "sin(150) = 1/2\r<BR>\n" );
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document.write( "sin(30) = sin(180 - 30)\r<BR>\n" );
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document.write( "30 - 180 would be - 150 degrees.\r<BR>\n" );
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document.write( "add 360 to it until it becomes positive and is between 0 and 360 degrees, and the equivalent angle is 210 degrees.\r<BR>\n" );
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document.write( "this angle is between 180 degrees and 270 degrees and is therefore in the third quadrant.\r<BR>\n" );
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document.write( "sin(210) = -1/2\r<BR>\n" );
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document.write( "the sine function is the same value except the sign of negative.\r<BR>\n" );
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document.write( "the following graph of the sine function and the value of y when x = 30 degrees, 150 degrees, and 210 degrees shows this to be true.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/061203.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the following pictures show the different angle as seen on the unit circle.\r<BR>\n" );
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document.write( "30 degrees is in the first quadrant where the sign of the sine function is positive.\r<BR>\n" );
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document.write( "150 degrees is in the second quadrant where the sign of the sine function is still positive.\r<BR>\n" );
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document.write( "210 degrees is in the third quadrant where the sign of the sine function is negative.\r<BR>\n" );
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document.write( "note that 150 degrees is the equivalent angle of 180 - 30 degrees and is in the second quadrant.\r<BR>\n" );
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document.write( "note also that 210 degrees is the equivalent angle of 30 - 180 = -150 degrees which becomes 210 degrees after adding 360 to it in order to make it the positive equivalent of -150 degrees.\r<BR>\n" );
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document.write( "to answer your question:\r<BR>\n" );
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document.write( "your solution is that sin(180 - x) is not the same as sin(x - 180)\r<BR>\n" );
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document.write( "here's the unit circle pictures.\r<BR>\n" );
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document.write( "in the pictures, the format shown is (a,b) where a is the cosine of the angle shown and b is the sine of the angle shown.\r<BR>\n" );
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document.write( "you can see that (sqrt(3)/2,1/3) for 30 degrees means cosine 30 = sqrt(3)/2 and sine 30 = 1/2.\r<BR>\n" );
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document.write( "similarly cosine 150 degrees is - sqrt(3)/2 and sine 150 degrees is 1/2.\r<BR>\n" );
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document.write( "similarly cosine 210 degrees is -sqrt(3)/2 and sine 210 degrees is -1/2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/061204.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/061205.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/061206.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the sine function is positive in quadrants 1 and 2, and negative in quadrants 3 and 4.\r<BR>\n" );
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document.write( "the cosine function is positive in quadrants 1 and 4, and negative in quadrants 2 and 3.\r<BR>\n" );
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document.write( "some references you might find helpful.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/geometry/unit-circle.html\" target = \"_blank\">https://www.mathsisfun.com/geometry/unit-circle.html</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.themathpage.com/atrig/unit-circle.htm\" target = \"_blank\">http://www.themathpage.com/atrig/unit-circle.htm</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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