document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=978511>Question 978511</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if sin(x)=1/3 and sec(y)=5/4, where 0 < x < pi/2 and 0 < y < pi/2, evaluate the expression cos (x-y), these < have the line under them so like equal to as well  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #600145 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=600145\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sin(x) = 1/3<BR>\n" );
document.write( "sec(y) = 5/4.\r<BR>\n" );
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document.write( "cos(x-y) = cos(x)cos(y) + sin(x)sin(y)\r<BR>\n" );
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document.write( "this is found by looking up trig identities, of which cos(x-y) or cos(a-b) is one of them.\r<BR>\n" );
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document.write( "to solve, you need sin and cos of x and you need sin and cos of y.\r<BR>\n" );
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document.write( "sin(x) = 1/3.<BR>\n" );
document.write( "1 = opposite<BR>\n" );
document.write( "3 = hypotenuse.<BR>\n" );
document.write( "by pythagorus:<BR>\n" );
document.write( "o^2 + a^2 = h^2<BR>\n" );
document.write( "this is short for opposite squared plus acjacent suared = hypotenuse squared.<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "1^2 + a^2 = 3^2 which becomes:<BR>\n" );
document.write( "1 + a^2 = 9 which becomes:<BR>\n" );
document.write( "a^2 = 8 which becomes:<BR>\n" );
document.write( "a = sqrt(8).\r<BR>\n" );
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document.write( "you have:<BR>\n" );
document.write( "sin(x) = 1/3<BR>\n" );
document.write( "cos(x) = sqrt(8)/3\r<BR>\n" );
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document.write( "sec(y) = 5/4<BR>\n" );
document.write( "since sec = 1/cos, this becomes:<BR>\n" );
document.write( "1/cos(y) = 5/4 which becomes:<BR>\n" );
document.write( "cos(y) = 4/5<BR>\n" );
document.write( "by pythagorus:<BR>\n" );
document.write( "o^2 + a^2 = h^2 which becomes:<BR>\n" );
document.write( "o^2 + 4^2 = 5^2 which becomes:<BR>\n" );
document.write( "o^2 + 16 = 25 which becomes:<BR>\n" );
document.write( "o^2 = 25 - 16 = 9 which becomes:<BR>\n" );
document.write( "o = 3\r<BR>\n" );
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document.write( "you have:\r<BR>\n" );
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document.write( "sin(x) = 1/3<BR>\n" );
document.write( "cos(x) = sqrt(8)/3<BR>\n" );
document.write( "sin(y) = 3/5<BR>\n" );
document.write( "cos(y) = 4/5\r<BR>\n" );
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document.write( "you can now solve the identity of cos(x-y) = cos(x)cos(y) + sin(x)sin(y), which becomes:\r<BR>\n" );
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document.write( "cos(x-y) = sqrt(8)/3 * 4/5 + 1/3 * 3/5 which becomes:\r<BR>\n" );
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document.write( "cos(x-y) = .954247\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "you can confirm by solving for each of the angles separately.\r<BR>\n" );
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document.write( "sin(x) = 1/3 gets you x = 19.471221\r<BR>\n" );
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document.write( "sin(y) = 3/5 gets you y = 36.869898\r<BR>\n" );
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document.write( "(x-y) gets you 19.471221 - 36.869898 = -17.398677 degrees.\r<BR>\n" );
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document.write( "cos(x-y) = cos(-17.398677) = .954245\r<BR>\n" );
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document.write( "answers are the same either way so it looks like you did good.\r<BR>\n" );
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