document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=967183>Question 967183</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find sin 2x, cos 2x, and tan 2x from the given information.<BR>\n" );
document.write( "sec x = 4,   x in Quadrant IV\r<BR>\n" );
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document.write( "sin 2x	=	<BR>\n" );
document.write( "cos 2x	=	<BR>\n" );
document.write( "tan 2x	=	 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #591171 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=591171\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you are given that sec x = 4 in quadrant 4.\r<BR>\n" );
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document.write( "secant is equal to 1 / cosine, so cos x = 1/4.\r<BR>\n" );
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document.write( "cosine is positive in quadrant 4 so this is good so far.\r<BR>\n" );
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document.write( "there's an easy way to do this and there's a hard way to do this.\r<BR>\n" );
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document.write( "here's the easy way.\r<BR>\n" );
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document.write( "cosine x = 1/4 means that x = 75.52248781 degrees.\r<BR>\n" );
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document.write( "that would be in quadrant 1.\r<BR>\n" );
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document.write( "in quadrant 4, x would be equal to 360 - 75.52248781 degrees which makes x equal to 284.4775122.\r<BR>\n" );
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document.write( "2x is therefore equal to 568.9550244 degrees.\r<BR>\n" );
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document.write( "you want to get the equivalent number of degrees between 0 and 360 degrees.\r<BR>\n" );
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document.write( "subtract 360 from 568.9550244 to get 208.9550244.\r<BR>\n" );
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document.write( "that's equal to 2x.\r<BR>\n" );
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document.write( "sin 2x is therefore equal to -.4841229183.<BR>\n" );
document.write( "cos 2x is therefore equal to -.875<BR>\n" );
document.write( "tan 2x is therefore equal to .5532833352.\r<BR>\n" );
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document.write( "the angle of x is in the fourth quadrant.<BR>\n" );
document.write( "the angle of 2x is in the third quadrant.\r<BR>\n" );
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document.write( "that's the easy way.\r<BR>\n" );
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document.write( "now we'll do it the hard way.<BR>\n" );
document.write( "we should get the same answer or we did it wrong.\r<BR>\n" );
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document.write( "you have sec x = 4 and x is in the fourth quadrant.\r<BR>\n" );
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document.write( "since sec x is equal to 1 / cos x, then 1 / cos x = 4 which makes cos x = 1/4.\r<BR>\n" );
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document.write( "since cosine is adjacent / hypotenuse, then you get adjacent is equal to 1 and hyupotenuse is equal to 4.\r<BR>\n" );
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document.write( "use pythagorus to find opposite.\r<BR>\n" );
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document.write( "pythagorus says adjacent squared plus opposite squared = hypotenuse squared.\r<BR>\n" );
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document.write( "this becomes 1 squared + opposite squared = 4 squared.\r<BR>\n" );
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document.write( "this becomes 1 + opposite squared = 16\r<BR>\n" );
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document.write( "solve for opposite squared to get opposite squared = 15.\r<BR>\n" );
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document.write( "take the square root of both sides of the equation to get opposite = sqrt(15).\r<BR>\n" );
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document.write( "you now have:\r<BR>\n" );
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document.write( "adjacent = 1<BR>\n" );
document.write( "opposite = sqrt(15)<BR>\n" );
document.write( "hypotenuse = 4.\r<BR>\n" );
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document.write( "from this, you can derive:\r<BR>\n" );
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document.write( "sin x = sqrt(15)/4<BR>\n" );
document.write( "cos x = 1/4<BR>\n" );
document.write( "tan x = sqrt(15)\r<BR>\n" );
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document.write( "this would be if the angle was in the first quadrant because all the functions are positive in the first quadrant.\r<BR>\n" );
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document.write( "since the angle is in the fourth quadrant, then sine is negative and tangent is negative while cosine is positive.\r<BR>\n" );
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document.write( "we therefore get:\r<BR>\n" );
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document.write( "sin x = -sqrt(15)/4<BR>\n" );
document.write( "cos x = 1/4<BR>\n" );
document.write( "tan x = -sqrt(15)\r<BR>\n" );
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document.write( "those are the trig functions in the fourth quadrant where our angle x resides.\r<BR>\n" );
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document.write( "now you want to find:\r<BR>\n" );
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document.write( "sin 2x =<BR>\n" );
document.write( "cos 2x =<BR>\n" );
document.write( "tan 2x = \r<BR>\n" );
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document.write( "the trig identity formulas for these are:\r<BR>\n" );
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document.write( "sin(2x) = 2 * sin(x) * cos(x)<BR>\n" );
document.write( "cos(2x) = cos^2(x) - sin^2(x)<BR>\n" );
document.write( "tan(2x) = 2 * tan(x) / (1 - tan^2(x))\r<BR>\n" );
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document.write( "using what we know of sin(x) and cos(x) and tan(x), we can solve these equations.\r<BR>\n" );
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document.write( "we know that:\r<BR>\n" );
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document.write( "sin x = -sqrt(15)/4<BR>\n" );
document.write( "cos x = 1/4<BR>\n" );
document.write( "tan x = -sqrt(15)\r<BR>\n" );
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document.write( "sin(2x) = 2 * -sqrt(15) / 4 * 1 / 4 = -sqrt(15) / 8 = -.4841229183.<BR>\n" );
document.write( "this agrees with what we found the easy way.\r<BR>\n" );
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document.write( "cos(2x) = cos^2(x) - sin^2(x) = (1/4)^2 - (-sqrt(15) / 4)^2 = -.875<BR>\n" );
document.write( "this also agrees with what we found the easy way.\r<BR>\n" );
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document.write( "tan(2x) = 2 * tan(x) / (1 - tan^2(x)) = 2 * -sqrt(15) / (1 - (-sqrt(15))^2) = .5532833352.<BR>\n" );
document.write( "this also agrees with what we found the easy way.\r<BR>\n" );
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document.write( "note on the easy way.<BR>\n" );
document.write( "it's not as easy as you might think.<BR>\n" );
document.write( "you have to know exactly what you are doing or you can screw it up easily.\r<BR>\n" );
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document.write( "you were probably meant to solve it the hard way which is thorugh the use of the double angle identity formulas.\r<BR>\n" );
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document.write( "i use the easy way to test whether my answer agrees with what i got from the identity formulas.\r<BR>\n" );
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document.write( "it does in this case so i'm reasonably confident the solution is good.\r<BR>\n" );
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document.write( "the easy way also helps to understand exactly what is happening and why these identity formula work.\r<BR>\n" );
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document.write( "but, once again, the easy way is not that easy and don't bother trying to learn it if you don't have to.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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