document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158099>Question 1158099</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me answer this question:\r<BR>\n" );
document.write( "\n" );
document.write( "Simplify<BR>\n" );
document.write( "sin (  -  θ) sin (90θ + θ)  -  cos (180 θ  -  θ) cos (270 θ +θ). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854568 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5377)</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=854568\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I believe there was a mistake copying the formula in the question.<BR>\n" );
document.write( "As written in the question, it says<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-theta%29sin%2890theta%2Btheta%29-cos%28180theta-theta%29cos%28270theta%2Btheta%29 ALIGN=MIDDLE  ALT=\"sin%28-theta%29sin%2890theta%2Btheta%29-cos%28180theta-theta%29cos%28270theta%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which could be written more simply as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-theta%29sin%2891theta%29-cos%28179theta%29cos%28271theta%29 ALIGN=MIDDLE  ALT=\"sin%28-theta%29sin%2891theta%29-cos%28179theta%29cos%28271theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I do not know of a way to further simplify that, and it does not sound like a typical high school math question.<BR>\n" );
document.write( " <BR>\n" );
document.write( "I suspect that the expression to simplify was meant to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-theta%29%2Asin%2890%5Eo%2Btheta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29 ALIGN=MIDDLE  ALT=\"sin%28-theta%29%2Asin%2890%5Eo%2Btheta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That could  be simplified using the relations know between the trigonometric functions sine and cosine of angles that are reflected or turned by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You could find those relations in any list of \"trigonometric identities\",<BR>\n" );
document.write( "but no need to memorize them, because you can visualize them for any angle in the unit circle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For example, if we change the sign of the angle , sine changes sign, but cosine stays the same:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sin%28-theta%29=-sin%28theta%29%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%28-theta%29=-sin%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28-theta%29=cos%28theta%29%29 ALIGN=MIDDLE  ALT=\"cos%28-theta%29=cos%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C300%2C-0.1%2C1.4%2C-0.9%2C1.35%2C%0D%0Aarrow%28-0.1%2C0%2C1.35%2C0%29%2Carrow%280%2C0%2C0%2C1.3%29%2Carrow%280%2C0%2C0%2C-1.3%29%2C%0D%0Acircle%280%2C0%2C1%29%2Ctriangle%280%2C0%2C0.92%2C0.39%2C0.92%2C-0.39%29%2C%0D%0Arectangle%280.92%2C-0.05%2C0.87%2C0.05%29%2Carc%280%2C0%2C0.6%2C0.6%2C0%2C22.9%29%2C%0D%0Aarc%280%2C0%2C0.7%2C0.7%2C-22.9%2C0%29%2Clocate%280.35%2C-0.03%2Ctheta%29%2C%0D%0Alocate%281.37%2C0%2Cx%29%2Clocate%280.05%2C1.35%2Cy%29%2Cgreen%28line%280.92%2C-0.02%2C0.92%2C-0.39%29%29%2C%0D%0Ared%28line%280%2C0%2C0.92%2C0%29%29%2Cgreen%28line%280.92%2C0.02%2C0.92%2C0.39%29%29%2C%0D%0Alocate%280.92%2C0.5%2CP%29%2Clocate%280.92%2C-0.39%2CQ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28red%28cos%28theta%29%29%2Cgreen%28sin%28theta%29%29%29 ALIGN=MIDDLE  ALT=\"P%28red%28cos%28theta%29%29%2Cgreen%28sin%28theta%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28red%28cos%28theta%29%29%2Cgreen%28sin%28-theta%29%29%29 ALIGN=MIDDLE  ALT=\"Q%28red%28cos%28theta%29%29%2Cgreen%28sin%28-theta%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( "Adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > to an angle is turning it 1/4 of a circle, and turns sine into cosine and cosine into -sine.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C200%2C-1.1%2C1.15%2C-0.1%2C1.4%2C%0D%0Aarrow%28-0.1%2C0%2C1.15%2C0%29%2Carrow%280%2C0%2C0%2C1.3%29%2Carrow%280%2C0%2C-1.3%2C0%29%2C%0D%0Acircle%280%2C0%2C1%29%2Ctriangle%280%2C0%2C0.92%2C0.39%2C0.92%2C-0%29%2C%0D%0Arectangle%280.92%2C0%2C0.87%2C0.05%29%2Carc%280%2C0%2C0.6%2C0.6%2C-112.9%2C-90%29%2C%0D%0Aarc%280%2C0%2C0.7%2C0.7%2C-22.9%2C0%29%2Clocate%280.35%2C0.15%2Ctheta%29%2C%0D%0Alocate%281.13%2C0%2Cx%29%2Clocate%280.05%2C1.35%2Cy%29%2C%0D%0Ared%28line%280%2C0%2C0.92%2C0%29%29%2Cgreen%28line%280.92%2C0.02%2C0.92%2C0.39%29%29%2C%0D%0Alocate%280.92%2C0.5%2CP%29%2C%0D%0Atriangle%280%2C0%2C-0.39%2C0.92%2C0%2C0.92%29%2Crectangle%280%2C0.92%2C-0.05%2C0.87%29%2C%0D%0Agreen%28line%28-0.39%2C0.92%2C-0.02%2C0.92%29%29%2Cred%28line%280%2C0%2C0%2C0.92%29%29%2C%0D%0Alocate%28-0.39%2C1.05%2CQ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28red%28cos%28theta%29%29%2Cgreen%28sin%28theta%29%29%29 ALIGN=MIDDLE  ALT=\"P%28red%28cos%28theta%29%29%2Cgreen%28sin%28theta%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%28green%28cos%2890%5Eo%2Btheta%29%29%2Cred%28sin%2890%5Eo%2Btheta%29%29%29 ALIGN=MIDDLE  ALT=\"Q%28green%28cos%2890%5Eo%2Btheta%29%29%2Cred%28sin%2890%5Eo%2Btheta%29%29%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=highlight%28sin%2890%5Eo%2Btheta%29=cos%28theta%29%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%2890%5Eo%2Btheta%29=cos%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2890%5Eo%2Btheta%29=-sin%28theta%29 ALIGN=MIDDLE  ALT=\"cos%2890%5Eo%2Btheta%29=-sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using the two highlighted relations above we can start simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-theta%29%2Asin%2890%5Eo%2Btheta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29 ALIGN=MIDDLE  ALT=\"sin%28-theta%29%2Asin%2890%5Eo%2Btheta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%28theta%29%2Acos%28theta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29 ALIGN=MIDDLE  ALT=\"-sin%28theta%29%2Acos%28theta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >}<BR>\n" );
document.write( " <BR>\n" );
document.write( "Also, adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > to an angle (turning it half a circle), causes the sign to change for the sine and cosine functions.<BR>\n" );
document.write( "So, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28180%5Eo-theta%29=-cos%28-theta%29 ALIGN=MIDDLE  ALT=\"cos%28180%5Eo-theta%29=-cos%28-theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28-theta%29=cos%28theta%29 ALIGN=MIDDLE  ALT=\"cos%28-theta%29=cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cos%28180%5Eo-theta%29=-cos%28theta%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%28180%5Eo-theta%29=-cos%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similarly<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28270%5Eo%2Btheta%29=cos%28180%5Eo%2B%2890%5Eo%2Btheta%29%29=-cos%2890%5Eo%2Btheta%29 ALIGN=MIDDLE  ALT=\"cos%28270%5Eo%2Btheta%29=cos%28180%5Eo%2B%2890%5Eo%2Btheta%29%29=-cos%2890%5Eo%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2890%5Eo%2Btheta%29=-sin%28theta%29 ALIGN=MIDDLE  ALT=\"cos%2890%5Eo%2Btheta%29=-sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cos%28270%5Eo%2Btheta%29=sin%28theta%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%28270%5Eo%2Btheta%29=sin%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Using the highlighted relations above we can continue simplifying:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28-theta%29%2Asin%2890%5Eo%2Btheta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29 ALIGN=MIDDLE  ALT=\"sin%28-theta%29%2Asin%2890%5Eo%2Btheta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%28theta%29%2Acos%28theta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29 ALIGN=MIDDLE  ALT=\"-sin%28theta%29%2Acos%28theta%29-cos%28180%5Eo-theta%29%2Acos%28270%5Eo%2Btheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >}<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%28theta%29%2Acos%28theta%29-%28-cos%2890%5Eo%2Btheta%29%29%2A%28sin%28theta%29%29 ALIGN=MIDDLE  ALT=\"-sin%28theta%29%2Acos%28theta%29-%28-cos%2890%5Eo%2Btheta%29%29%2A%28sin%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sin%28theta%29%2Acos%28theta%29%2Bsin%28theta%29%2Acos%28theta%29=highlight%280%29 ALIGN=MIDDLE  ALT=\"-sin%28theta%29%2Acos%28theta%29%2Bsin%28theta%29%2Acos%28theta%29=highlight%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );