document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=991180>Question 991180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Why does sin 30° = 150° (or sin &#960;/6 = sin 5&#960;/6)? Refer to both the unit <BR>\n" );
document.write( "circle and the graph of the sine curve. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #611190 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=611190\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sine (30) = .5<BR>\n" );
document.write( "sine (150) = .5<BR>\n" );
document.write( "why are they the same.<BR>\n" );
document.write( "it's because the reference angle for 150 is equal to 30.\r<BR>\n" );
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document.write( "that reference angle is the angle within the triangle formed from dropping a perpendicular to the x-axis of the unit circle.\r<BR>\n" );
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document.write( "here's a sketch of 30 degrees and 150 degrees on the unit circle and the triangles formed.\r<BR>\n" );
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document.write( "the 30 degree angle is in quadrant 1.\r<BR>\n" );
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document.write( "the 150 degree angle is in quadrant 2.\r<BR>\n" );
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document.write( "the internal angle of the triangle in quadrant 2 is equal to 180 - 150 which is equal to 30 degrees.\r<BR>\n" );
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document.write( "that is called the reference angle.\r<BR>\n" );
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document.write( "it is equivalent to the same angle in quadrant 1.\r<BR>\n" );
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document.write( "the sine of 30 degrees is equal to .5\r<BR>\n" );
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document.write( "the sine of 150 degrees is equal to .5\r<BR>\n" );
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document.write( "you can confirm using your calculator.\r<BR>\n" );
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document.write( "the graph of sin(x) and the intersection of that graph with sin(30) is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/092204.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the angles of interest to you are 30 and 150.\r<BR>\n" );
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document.write( "you can see that the sine is .5 for both of them.\r<BR>\n" );
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document.write( "the other two angles are in quadrant 3 and quadrant 4.\r<BR>\n" );
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document.write( "the sine is the same except the sine is negative.\r<BR>\n" );
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document.write( "that's because sine is positive in quadrants 1 and 2 and is negative in quadrants 3 and 4.\r<BR>\n" );
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document.write( "on the unit circle, sine is equal to y / 1.\r<BR>\n" );
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document.write( "in quadrants 1 and 2, y is positive.\r<BR>\n" );
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document.write( "in quadrants 3 and 4, y is negative.\r<BR>\n" );
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document.write( "a 30 degree angle in quadrant 2 is 180 - 30 = 150.<BR>\n" );
document.write( "a 30 degree angle in quadrant 3 is 180 + 30 = 210.<BR>\n" );
document.write( "a 30 degree angle in quadrant 4 is 360 - 30 = 330.\r<BR>\n" );
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document.write( "if you are given the larger angles, you find the reference angle as follows:\r<BR>\n" );
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document.write( "in quadrant 2, the reference angle is 180 - 150 = 30.<BR>\n" );
document.write( "in quadrant 3, the reference angle is 210 - 180 = 30.<BR>\n" );
document.write( "in quadrant 4, the reference angle is 360 - 330 = 30.\r<BR>\n" );
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