document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=871093>Question 871093</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the exact value of the expression.\r<BR>\n" );
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document.write( "sin(cos^-1 2/3 - tan^-1 1/4) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #525332 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=525332\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> NOTE: I am rushing to get done before I have to go to my day job, so check for mistakes.<BR>\n" );
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document.write( "In this problem \"cos^-1\" means the inverse function of cosine,<BR>\n" );
document.write( "and \"tan^-1\" means the inverse function of tangent.<BR>\n" );
document.write( "We are looking for the exact value of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A-B%29 ALIGN=MIDDLE  ALT=\"sin%28A-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > knowing that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28A%29=2%2F3 ALIGN=MIDDLE  ALT=\"cos%28A%29=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28B%29=1%2F4 ALIGN=MIDDLE  ALT=\"tan%28B%29=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and that those inverse functions are defined so that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be in quadrants I or II, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be in quadrants I or IV.<BR>\n" );
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document.write( "To calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A-B%29 ALIGN=MIDDLE  ALT=\"sin%28A-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can use the trigonometric identity<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A-B%29=sin%28A%29%2Acos%28B%29-sin%28B%29%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"sin%28A-B%29=sin%28A%29%2Acos%28B%29-sin%28B%29%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we need to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29 ALIGN=MIDDLE  ALT=\"sin%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29 ALIGN=MIDDLE  ALT=\"sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28B%29 ALIGN=MIDDLE  ALT=\"cos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "In quadrants I and I <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29%3E0 ALIGN=MIDDLE  ALT=\"sin%28A%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29=sqrt%281-%28cos%28A%29%29%5E2%29=sqrt%281-%282%2F3%29%5E2%29=sqrt%281-4%2F9%29=sqrt%285%2F9%29=sqrt%285%29%2F9 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28B%29=1%2F4%3E0 ALIGN=MIDDLE  ALT=\"tan%28B%29=1%2F4%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > only hapens in quadrants I and III,<BR>\n" );
document.write( "and the definition of the inverse tangent function says that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in I or IV,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is definitely in quadrant I, where sine and cosine are positive.<BR>\n" );
document.write( "This is how angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > would look in a right triangle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C100%2C-0.25%2C4.25%2C-0.25%2C1.25%2C%0D%0Atriangle%280%2C0%2C4%2C0%2C4%2C1%29%2C%0D%0Alocate%28-0.2%2C0.1%2CB%29%2Clocate%281.8%2C0%2C4%29%2Clocate%284.05%2C0.7%2C1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The hypotenuse of that right triangle measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%5E2%2B1%5E2%29=sqrt%2816%2B1%29=sqrt%2817%29 ALIGN=MIDDLE  ALT=\"sqrt%284%5E2%2B1%5E2%29=sqrt%2816%2B1%29=sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29=1%2Fsqrt%2817%29 ALIGN=MIDDLE  ALT=\"sin%28B%29=1%2Fsqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28B%29=4%2Fsqrt%2817%29 ALIGN=MIDDLE  ALT=\"cos%28B%29=4%2Fsqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting all the values found into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A-B%29=sin%28A%29%2Acos%28B%29-sin%28B%29%2Acos%28A%29 ALIGN=MIDDLE  ALT=\"sin%28A-B%29=sin%28A%29%2Acos%28B%29-sin%28B%29%2Acos%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A-B%29=%28sqrt%285%29%2F9%29%2A%284%2Fsqrt%2817%29%29-%281%2Fsqrt%2817%29%29%2A%282%2F3%29=4sqrt%285%29%2F%289sqrt%2817%29%29-2%2F%283sqrt%2817%29%29=4sqrt%285%29%2F%289sqrt%2817%29%29-6%2F%289sqrt%2817%29%29=%284sqrt%285%29-6%29%2F%289sqrt%2817%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "SInce that does not look elegant, and teachers do not like square roots in denominators, we multiply numerator and denominator times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2817%29 ALIGN=MIDDLE  ALT=\"sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284sqrt%285%29sqrt%2817%29-6sqrt%2817%29%29%2F%289%2A17%29=%284sqrt%2885%29-6sqrt%2817%29%29%2F153 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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