document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1009209>Question 1009209</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two polar coordinate representations for the rectangular coordinate point <BR>\n" );
document.write( "(&#8722;6, 2 sqrt 3), one with r > 0 and one with r < 0 and both with 0 &#8804; &#952; < 2&#960;.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #624758 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=624758\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the rectangular coordinate is (-6,2sqrt(3))\r<BR>\n" );
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document.write( "this point is in the second quadrant where the x-coordinate is -6 and the y-coordinate is 2sqrt(3).\r<BR>\n" );
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document.write( "this creates a triangle where the length of the adjacent side is 6 units in length and the opposite side is 2sqrt(3) units in length.\r<BR>\n" );
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document.write( "to find the angle in the triangle we find tan^-1(2sqrt(3)/6) which is equal to 30 degrees.\r<BR>\n" );
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document.write( "that angle is the reference angle.\r<BR>\n" );
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document.write( "the reference angle is the equivalent angle in the first quadrant.\r<BR>\n" );
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document.write( "to find the angle in the second quadrant, we take 180 - 30 = 150 degrees.\r<BR>\n" );
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document.write( "the angle we want is 150 degrees.\r<BR>\n" );
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document.write( "the length of the hypotenuse of this triangle is equal to sqrt(48).\r<BR>\n" );
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document.write( "that's by pythagorus that says that x^2 + y^2 = h^2.\r<BR>\n" );
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document.write( "we get (2sqrt(3))^2 + (6)^2 = 12 + 36 = 48 which makes h = sqrt(48).\r<BR>\n" );
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document.write( "h = hypotenuse of the right triangle and also equals r.\r<BR>\n" );
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document.write( "the point (-6,2sqrt(3)) in rectnagular coordinates is the same as the point (sqrt(48),150 degrees) in polar coordinates.\r<BR>\n" );
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document.write( "so one way to describe the point is (sqrt(48),150 degrees)\r<BR>\n" );
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document.write( "this is the form of (r,angle), where r = sqrt(48) and angle = 150 degrees.\r<BR>\n" );
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document.write( "another way to describe the point is (-sqrt(48),330 degrees).\r<BR>\n" );
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document.write( "we get this by adding 180 to 150 to get 330 and then taking -r rather than r.\r<BR>\n" );
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document.write( "it's another way of describing the exact same point.\r<BR>\n" );
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document.write( "the point (-6,2sqrt(3)) in rectangular coordinates is equivalent to (sqrt(48),150 degrees) or (-sqrt(48),330 degrees) in polar coordinates.\r<BR>\n" );
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document.write( "to convert from degrees to radians, multiply degrees by pi and then divide by 180 to get radians.\r<BR>\n" );
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document.write( "150 degrees * pi / 180 is equal to 5/6 * pi radians.\r<BR>\n" );
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document.write( "330 degrees * pi / 180 is equal to 11/6 * pi radians.\r<BR>\n" );
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document.write( "in radians, then:\r<BR>\n" );
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document.write( "(-6,2sqrt(3)) equals (sqrt(48),5/6*pi radians) and (-sqrt(48),11/6*pi radians).\r<BR>\n" );
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document.write( "visually, this looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/120909.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "both of these diagrams point to the same point of (-6,2sqrt(3)).\r<BR>\n" );
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document.write( "when r is minus, you go the indicated number of units in the opposite direction shown.\r<BR>\n" );
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document.write( "here's a reference to some good lessons on polar cooraindates by professor robbob.\r<BR>\n" );
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document.write( "there are 3 videos in the series.\r<BR>\n" );
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document.write( "take a look at the first and it should then point to the second and third if you wish to learn further.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= https://www.youtube.com/watch?v=tKi05dfUhAA\"\" target = \"_blank\">https://www.youtube.com/watch?v=tKi05dfUhAA</a>\r<BR>\n" );
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