document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=427310>Question 427310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1/2cosx cannot equal cos1/2X<BR>\n" );
document.write( "Discuss how the 1/2 part affects the graphs of y=1/2cosx and y=cos1/2x\r<BR>\n" );
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document.write( "I dont understand what each 1/2 does.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #297182 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=297182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (1/2) * cosine (x) is not the same as cosine ((1/2) * x).\r<BR>\n" );
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document.write( "1/2 * cosine (x) means you calculate the cosine of x and then you divide it by 2.\r<BR>\n" );
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document.write( "cosine ((1/2) * x) means you are dividing the angle by 2 and then taking the cosine of that.\r<BR>\n" );
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document.write( "assuming your angle is 60 degrees, then:\r<BR>\n" );
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document.write( "cosine (60) = .5\r<BR>\n" );
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document.write( "(1/2) * cosine (60) = (1/2) * .5 = .25\r<BR>\n" );
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document.write( "cosine ((1/2) * 60) = cosine (30) = .866025404\r<BR>\n" );
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document.write( "you can see the difference by looking at the triangles formed on the unit circle.\r<BR>\n" );
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document.write( "the picture of the unit circle is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/unit_circle_20110326_1.jpg\" alt = \"***** picture not found *****\" />\r<BR>\n" );
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document.write( "in the unit circle, the 60 degree angle forms triangle ADE.\r<BR>\n" );
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document.write( "the cosine of 60 degrees would be equal to adjacent / hypotenuse which is equal to AE / AD.\r<BR>\n" );
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document.write( "since the hypotenuse of the triangles formed in the unit circle are always equal to 1, this means that the cosine of 60 degrees is equal to AE / AD which is equal to AE / 1 which is equal to AE.\r<BR>\n" );
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document.write( "half the cosine of 60 degrees would be half the length of AE.\r<BR>\n" );
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document.write( "when you half the angle of 60 degrees, you get the angle of 30 degrees.\r<BR>\n" );
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document.write( "the angle of 30 degrees on the unit circle forms triangle ABC.\r<BR>\n" );
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document.write( "the cosine of 30 degrees is equal to adjacent / hypotenuse which is equal to AC / AB.\r<BR>\n" );
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document.write( "since the hypotenuse of the triangles formed in the unit circle are always equal to 1, then cosine of 30 degrees equals AC / AB which is equal to AC / 1 which is equal to AC.\r<BR>\n" );
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document.write( "you have 1/2 the cosine of 60 degrees equal to 1/2 the length of AE and you have the cosine of 1/2 of 60 degrees equal to the cosine of 30 degrees equal to the length of AC.\r<BR>\n" );
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document.write( "that's the difference between 1/2 the cosine of an angle and the cosine of 1/2 of the angle.\r<BR>\n" );
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document.write( "hopefully that makes sense to you.\r<BR>\n" );
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document.write( "if not, let me know where you are still confused and i'll try to clear it up.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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