document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206406>Question 1206406</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with this equation.<BR>\n" );
document.write( "Find sin(2x), cos(2x), and tan(2x) from the given information.<BR>\n" );
document.write( "tan(x) = −4/3, x in Quadrant II<BR>\n" );
document.write( "sin(2x)	=	<BR>\n" );
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document.write( "cos(2x)	=	<BR>\n" );
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document.write( "tan(2x)	=	<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 #843845 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=843845\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you are given that tan(x) = -4/3 and x is in the second quadrant.\r<BR>\n" );
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document.write( "tan is opposite divided by adjacent.<BR>\n" );
document.write( "that makes opposite = 4 and adjacent = -3<BR>\n" );
document.write( "hypotenuse = sqrt(4^2 + (-3)^2) = 5.\r<BR>\n" );
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document.write( "in the graph of angles, x represents the side adjacent to the angle and y represents the side opposite the angle.\r<BR>\n" );
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document.write( "in the second quadrant, x is minus and y is positive.<BR>\n" );
document.write( "tan(x) = -4/3 is opposite / adjacent is y/x.<BR>\n" );
document.write( "y is positive and x is negative, so -(4/3) is 4/-3, and not -3/4.<BR>\n" );
document.write( "since -4/3 is equivalent to 4/-3 is equivalent to -(4/3), you can use either one if the formula allows it.\r<BR>\n" );
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document.write( "the simplest way to find your answer is to use the double angle identity formulas.\r<BR>\n" );
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document.write( "tan(2x) = 2tan(x) / (1-tan^2(x))<BR>\n" );
document.write( "since tan(x) = -4/3 in the second quadrant, you get:<BR>\n" );
document.write( "tan(2x) = 2 * -4/3) / (1 - (-4/3)^2) = 3.428571429.<BR>\n" );
document.write( "the answer in simplified fraction form would be 24/7.\r<BR>\n" );
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document.write( "sin(2x) = 2 * sin(x) * cos(x)<BR>\n" );
document.write( "since sin(x) = 4/5 in the second quadrant and cos(x) = -3/5 in the second quadrant, you get:<BR>\n" );
document.write( "sin(2x) = 2 * 4/5 * -3/5 = -.96<BR>\n" );
document.write( "the answer in simplified fraction form would be -24/25.\r<BR>\n" );
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document.write( "cos(2x) = cos^2(x) - sin^2(x)<BR>\n" );
document.write( "since sin(x) = 4/5 in the second quadrant and cos(x) = -3/5 in the second quadrant, you get:<BR>\n" );
document.write( "cos(2x) = (-3/5)^2 - (4/5)^2 = -.28<BR>\n" );
document.write( "the answer in simplified fraction form would be -7/25.\r<BR>\n" );
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document.write( "you can confirm these answers are good in the following manner.<BR>\n" );
document.write( "you are given that tan(x) = -4/3 in the second quadrant.<BR>\n" );
document.write( "use your calculator to solve for the angle to get angle = -53.13010235 degrees.<BR>\n" );
document.write( "convert that to a positive equivalent angle by adding 360 to it to get 306.8690976 degrees.<BR>\n" );
document.write( "that's in the fourth quadrant.<BR>\n" );
document.write( "the equivalent angle in the first quadrant is 360 minus that = 53.13010235 degrees.<BR>\n" );
document.write( "the equivalent angle in the second quadrant is 180 minus that = 126.8698976 degrees.\r<BR>\n" );
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document.write( "to confirm this is the correct angle, find tan(126.8698976) = -1.33333.....<BR>\n" );
document.write( "convert to fraction to get -4/3, which is correct because that's what was given.\r<BR>\n" );
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document.write( "you have x = 126.8698976 degrees.<BR>\n" );
document.write( "2x is therefore equal to 253.7397953 degrees.\r<BR>\n" );
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document.write( "sin (that) = -.96<BR>\n" );
document.write( "cos(that) = -.28<BR>\n" );
document.write( "tan(that) = 3.428571429.\r<BR>\n" );
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document.write( "these values agree with what we got earlier using the double angle formulas, so the answer is confirmed to be good.\r<BR>\n" );
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document.write( "the double angle identity formulas can be found in the following reference.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/List_of_trigonometric_identities\" target = \"_blank\">https://en.wikipedia.org/wiki/List_of_trigonometric_identities</a> under the heading of Multiple-angle and half-angle formulae\r<BR>\n" );
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