document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139919>Question 1139919</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If angle θ  is a principal angle that lies in quadrant 3 such that 0° ≤ θ ≤360°, determine the exact values of x, y and r for tan θ = 2/3  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760443 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=760443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> tan(theta) = 2/3\r<BR>\n" );
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document.write( "tan(theta) = opposite / adjacent.\r<BR>\n" );
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document.write( "this means that opposite is equal to 2 and adjacent is equal to 3.\r<BR>\n" );
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document.write( "since y is opposite the angle and x is adjacent the angle, then you get y = 2 and x = 3 in the first quadrant.\r<BR>\n" );
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document.write( "r is the hypotenuse of the right triangle formed.\r<BR>\n" );
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document.write( "the formula for r is r = sqrt(x^2 + y^2).\r<BR>\n" );
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document.write( "this makes r = sqrt(3^2 + 2^2) = sqrt(13).\r<BR>\n" );
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document.write( "in the third quadrant, y = -2 and x = -3.\r<BR>\n" );
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document.write( "tan(theta) = -2 / -3 = 2/3, same as in the first quadrant.\r<BR>\n" );
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document.write( "the value of the tangent function is the same whether or not the angle is in the first quadrant or in the third quadrant.\r<BR>\n" );
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document.write( "you can use your calculator to find theta.\r<BR>\n" );
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document.write( "arctan(2/3) = 33.69006753 degrees.\r<BR>\n" );
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document.write( "that's in the first quadrant.\r<BR>\n" );
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document.write( "that's also called the reference angle.\r<BR>\n" );
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document.write( "in the third quadrant, the equivalent angle would be 180 + 33.69006753 = 213.69006753 degrees.\r<BR>\n" );
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document.write( "the reference angle for 213.69006753 degrees is equal to 213.6906753 degrees minus 180 degrees = 33.6006753.\r<BR>\n" );
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document.write( "use your calculator to find tan(33.69006753) = .666666... = 2/3\r<BR>\n" );
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document.write( "use your calculator to find tan(213.69006753) = .66666.... = 2/3.\r<BR>\n" );
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document.write( "that means that 33.69006753 degrees and 213.69006753 degrees are equivalent angle because they have the same trigonometric values except for the signs.\r<BR>\n" );
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document.write( "sine(33.69006753) = .5547001962<BR>\n" );
document.write( "sine(213.69006753) = -.5547001962\r<BR>\n" );
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document.write( "same value for sine except for the sign.\r<BR>\n" );
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document.write( "cosine(33.6906753)) = .8320502943<BR>\n" );
document.write( "cosine(213.69006753) = -.8320502943\r<BR>\n" );
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document.write( "same value for cosine except for the sign.\r<BR>\n" );
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document.write( "we already did the tangent function.\r<BR>\n" );
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document.write( "secant is reciprocal of cosine, so the values will be the same except for the sign.\r<BR>\n" );
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document.write( "cosecant is reciprocal of sine, so the values will be the same except for the sign.\r<BR>\n" );
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document.write( "cotangent is reciprocal of tangent, so the values will be the same.\r<BR>\n" );
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document.write( "here's my picture of the angle in the first quadrant and the third quadrant.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/042612.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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