document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=889064>Question 889064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given tan theta = 8/5 and pi < theta < 3pi/2. Find the exact value of sin theta/2, cos theta/2, tan theta/2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #538207 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=538207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'll use x instead of theta since it's easier to type.\r<BR>\n" );
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document.write( "your problem statement becomes:\r<BR>\n" );
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document.write( "Given tan x = 8/5 and pi < x < 3pi/2. Find the exact value of sin x/2, cos x/2, tan x/2.\r<BR>\n" );
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document.write( "pi < x < 3pi/2.\r<BR>\n" );
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document.write( "in degrees, that means 180 < x < 270.\r<BR>\n" );
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document.write( "that puts x in the third quadrant.\r<BR>\n" );
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document.write( "tan(x) in the third quadrant is positive, but x and y in the third quadrant are negative, so tan(x) in the third quadrant is equal to -8/-5.\r<BR>\n" );
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document.write( "the value of x in the first quadrant is arctan(8/5) = 57.99461679 degrees.\r<BR>\n" );
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document.write( "the value of x in the third quarter is 180 + 57.99461679 degrees which is equal to 237.99461679 degrees.\r<BR>\n" );
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document.write( "you can test if this angle is correct by taking the tangent of it.\r<BR>\n" );
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document.write( "you will get tan(x) = 1.6 which is equivalent to 8/5, or more correctly, -8/-5.\r<BR>\n" );
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document.write( "half of 237.99461679 degrees is 118.9973084 degrees is equal to x/2.\r<BR>\n" );
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document.write( "118.9973084 degrees is in the second quadrant.\r<BR>\n" );
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document.write( "use your calculator to get:\r<BR>\n" );
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document.write( "sin(x/2) = .8746424812...<BR>\n" );
document.write( "cos(x/2) = -.4847685324...<BR>\n" );
document.write( "tan(x/2) = -1.804247642...\r<BR>\n" );
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document.write( "if my assumptions are correct, these are the values we should get when we use the half angle formulas.\r<BR>\n" );
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document.write( "the hypotenuse of the triangle containing the reference angle of x is equal to sqrt(8^2+5^2) = sqrt(64+25) = sqrt(89)\r<BR>\n" );
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document.write( "so our triangle has:\r<BR>\n" );
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document.write( "an opposite side from angle x of -8<BR>\n" );
document.write( "an adjacent side from angle x of -5<BR>\n" );
document.write( "a hypotenuse of sqrt(89)\r<BR>\n" );
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document.write( "we get:\r<BR>\n" );
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document.write( "sin(x) = -8/sqrt(89) = -.847998304...<BR>\n" );
document.write( "cos(x) = -5/sqrt(89) = -.52999894...<BR>\n" );
document.write( "tan(x) = -8/-5 = 1.6\r<BR>\n" );
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document.write( "the half angle identify formulas are:\r<BR>\n" );
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document.write( "sin(x/2) = +/- sqrt((1-cos(x))/2)\r<BR>\n" );
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document.write( "cos(x/2) =+/- sqrt((1+cos(x))/2)\r<BR>\n" );
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document.write( "tan(x/2) = +/- sqrt((1-cos(x))/(1+cos(x)))\r<BR>\n" );
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document.write( "it's safe to say that, if x is in the third quadrant, than x/2 is in the second quadrant.\r<BR>\n" );
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document.write( "a quick test reveals this to be true.\r<BR>\n" );
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document.write( "180/2 = 90 which is the lowest value that x/2 will take and is on the border between the first quadrant and the second quadrant.\r<BR>\n" );
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document.write( "270/2 = 135 which is in the middle of the second quadrant.\r<BR>\n" );
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document.write( "if x is in the third quadrant, x/2 has to be in the second quadrant.\r<BR>\n" );
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document.write( "now the formula for sin(x/2) is equal to +/- sqrt((1-cos(x))/2)\r<BR>\n" );
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document.write( "that becomes:\r<BR>\n" );
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document.write( "sin(x/2) = +/- .8746424812...\r<BR>\n" );
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document.write( "since sine is positive in the second quarter, then sin(x/2) = .8746424812.\r<BR>\n" );
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document.write( "this is the same as we got up above, so we are doing ok so far.\r<BR>\n" );
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document.write( "now the formula for cos(x/2) is equal to +/- sqrt((1+cos(x))/2)\r<BR>\n" );
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document.write( "that becomes:\r<BR>\n" );
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document.write( "cos(x/2) = +/- .4847685324...\r<BR>\n" );
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document.write( "since cosine is negative in the second quarter, then cos(x/2) = -.4847685324...\r<BR>\n" );
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document.write( "since this is the same as we got earlier, this is ok as well.\r<BR>\n" );
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document.write( "now the formula for tan(x/2) is equal to +/- sqrt((1-cos(x))/(1+cos(x))).\r<BR>\n" );
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document.write( "since tan is negative in the second quarter, we'll be looking for - sqrt... and not plus.\r<BR>\n" );
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document.write( "that becomes:\r<BR>\n" );
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document.write( "tan(x/2) = -1.804247642...\r<BR>\n" );
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document.write( "once again, this agrees with what we got above, so the assumption is that we did this correctly.\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "sin(x/2) = .8746424812...<BR>\n" );
document.write( "cos(x/2) = -.4847685324...<BR>\n" );
document.write( "tan(x/2) = -1.804247642...\r<BR>\n" );
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document.write( " </SPAN>\n" );
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