document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1125018>Question 1125018</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Using the graph of y=sin(x), list all values on the interval [-8pi/3, 13pi/6] that satisfy the ordered pair (x,1/2).<BR>\n" );
document.write( "List all the values that satisfy (x,1/2). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #741365 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=741365\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the angles are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-11pi/6, -7pi/6, pi/6, 5pi/6, 13pi/6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "first thing you do is find the angle in the first quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sin^-1(1/2) = pi/6 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply that by 180 / pi and it becomes 30 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this makes sense since sin(30 degrees) = 1/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sine is positive in the first and second quadrant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equivalent angle in the second quadrant is pi - pi/6 = 5pi/6 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's equivalent to 180 - 30 = 150 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5pi/6 * 180 / pi = 5/6 * 180 = 150 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that also makes sense.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "these angles will repeat every 360 degrees or every 2 pi radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2pi radians is equivalent to 12pi / 6 radians.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "pi / 6 + 12pi / 6 = 13pi / 6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's as far as you want to go in the positive direction.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so far your angles are pi/6, 5pi/6, 13pi/6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the other direction, pi/6 - 12pi / 6 = -11 pi/6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can't go down further than that because the limit is -16pi / 6 and subtracting another 12pi / 6 will go beyond that.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5pi / 6 - 12pi / 6 = -7pi / 6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can't go further than that because subtracting another 12pi / 6 will get you -17pi / 6 which is further than -16pi / 6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your angles are therefore:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-11pi/6, -7pi/6, pio/6, 5pi/6, 13pi/6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "all these angles will have 1/2 as their sine.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can use your calculator to confirm.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "alternatively, you can graph the equations to get what is shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/100207.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of y = sin(x) is in red.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graph of y = 1/2 is in purple.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the intersection of the red graph and and the purple graph shows the angles where sin(x) = 1/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the valid interval is between -16pi/6 and 13pi/6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the unshaded area of the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-16pi / 6 is the same angle as -8pi / 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "rather than simplifying all the angles, i keep the denominators the same.<BR>\n" );
document.write( "it's easier to see the intervals that way.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the graphing software will, however, do the simplification where it can.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );