document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207589>Question 1207589</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram, ABCD is a square.  Find sin angle PAQ.\r<BR>\n" );
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document.write( "P is on BC such that BP = 2 and PC = 3, and Q is on CD such that CQ = 2 and QD = 3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845546 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=845546\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "This is possibly what the diagram looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-3%2C3%2C-3%2C3%2C%0D%0A%0D%0Alocate%282%2C-1%2C%222%22%29%2Clocate%282%2C0.5%2C%223%22%29%2C%0D%0Alocate%28-0.7%2C2%2C%223%22%29%2Clocate%281.1%2C2%2C%222%22%29%2C%0D%0Alocate%280%2C-1.7%2C%225%22%29%2Clocate%28-1.7%2C0%2C%225%22%29%2C%0D%0A%0D%0Alocate%28-1.3-0.3%2C-1.46-0.2%2C%22A%22%29%2Clocate%281.7%2C-1.46-0.2%2C%22B%22%29%2Clocate%281.7%2B0.1%2C1.54%2C%22C%22%29%2Clocate%28-1.3-0.3%2C1.54%2B0.2%2C%22D%22%29%2Clocate%281.7%2B0.1%2C-0.46%2C%22P%22%29%2Clocate%280.7-0.2%2C1.54%2B0.3%2C%22Q%22%29%2C%0D%0A%0D%0Acircle%28-1.4%2C-1.56%2C0.05%29%2Ccircle%28-1.4%2C-1.56%2C0.07%29%2Ccircle%28-1.4%2C-1.56%2C0.09%29%2Ccircle%28-1.4%2C-1.56%2C0.11%29%2Ccircle%281.6%2C-1.56%2C0.05%29%2Ccircle%281.6%2C-1.56%2C0.07%29%2Ccircle%281.6%2C-1.56%2C0.09%29%2Ccircle%281.6%2C-1.56%2C0.11%29%2Ccircle%281.6%2C1.44%2C0.05%29%2Ccircle%281.6%2C1.44%2C0.07%29%2Ccircle%281.6%2C1.44%2C0.09%29%2Ccircle%281.6%2C1.44%2C0.11%29%2Ccircle%28-1.4%2C1.44%2C0.05%29%2Ccircle%28-1.4%2C1.44%2C0.07%29%2Ccircle%28-1.4%2C1.44%2C0.09%29%2Ccircle%28-1.4%2C1.44%2C0.11%29%2Ccircle%281.6%2C-0.56%2C0.05%29%2Ccircle%281.6%2C-0.56%2C0.07%29%2Ccircle%281.6%2C-0.56%2C0.09%29%2Ccircle%281.6%2C-0.56%2C0.11%29%2Ccircle%280.6%2C1.44%2C0.05%29%2Ccircle%280.6%2C1.44%2C0.07%29%2Ccircle%280.6%2C1.44%2C0.09%29%2Ccircle%280.6%2C1.44%2C0.11%29%2C%0D%0A%0D%0Aline%28-1.4%2C-1.56%2C1.6%2C-1.56%29%2Cline%281.6%2C-1.56%2C1.6%2C1.44%29%2Cline%281.6%2C1.44%2C-1.4%2C1.44%29%2Cline%28-1.4%2C1.44%2C-1.4%2C-1.56%29%2Cline%28-1.4%2C-1.56%2C0.6%2C1.44%29%2Cline%28-1.4%2C-1.56%2C1.6%2C-0.56%29%2C%0D%0A%0D%0Alocate%28-2.5%2C-2.5%2Cmatrix%281%2C4%2C%22Diagram%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Focus on right triangle PAB.<BR>\n" );
document.write( "We'll use the tangent trig ratio to determine the following.<BR>\n" );
document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(angle PAB) = PB/AB<BR>\n" );
document.write( "tan(angle PAB) = 2/5<BR>\n" );
document.write( "angle PAB = arctan(2/5)<BR>\n" );
document.write( "angle PAB = 21.801409 degrees approximately.\r<BR>\n" );
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document.write( "Now move your attention to right triangle DAQ.<BR>\n" );
document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(angle DAQ) = DQ/AD<BR>\n" );
document.write( "tan(angle DAQ) = 3/5<BR>\n" );
document.write( "angle DAQ = arctan(3/5)<BR>\n" );
document.write( "angle DAQ = 30.963757 degrees approximately.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "(angleDAQ) + (anglePAQ) + (anglePAB) = angleDAB<BR>\n" );
document.write( "(angleDAQ) + (anglePAQ) + (anglePAB) = 90<BR>\n" );
document.write( "anglePAQ = 90 - (angleDAQ + anglePAB)<BR>\n" );
document.write( "anglePAQ = 90 - (30.963757 + 21.801409)<BR>\n" );
document.write( "anglePAQ = 37.234834 degrees approximately<BR>\n" );
document.write( "sin(angle PAQ) = sin(37.234834) = <font color=red>0.605083 approximately</font>\r<BR>\n" );
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document.write( "------------------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Another approach.\r<BR>\n" );
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document.write( "Use the Pythagorean Theorem to determine these lengths<BR>\n" );
document.write( "QA = sqrt(5^2+3^2) = 5.830952 approximately<BR>\n" );
document.write( "PA = sqrt(5^2+2^2) = 5.385165 approximately<BR>\n" );
document.write( "QP = sqrt(2^2+3^2) = 3.605551 approximately\r<BR>\n" );
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document.write( "Focus on triangle PAQ of the diagram shown above.<BR>\n" );
document.write( "Use the Law of Cosines to find angle PAQ.<BR>\n" );
document.write( "a^2 = b^2+c^2-2*b*c*cos(A)<BR>\n" );
document.write( "(QP)^2 = (QA)^2+(PA)^2-2*(QA)*(PA)*cos(angle PAQ)<BR>\n" );
document.write( "(3.605551)^2 = (5.830952)^2+(5.385165)^2-2*(5.830952)*(5.385165)*cos(angle PAQ)<BR>\n" );
document.write( "12.999998 = 63.000003 -62.801277*cos(angle PAQ) <BR>\n" );
document.write( "12.999998 - 63.000003 = -62.801277cos(angle PAQ)<BR>\n" );
document.write( "-50.000005 = -62.801277cos(angle PAQ)<BR>\n" );
document.write( "cos(angle PAQ) = -50.000005/(-62.801277)<BR>\n" );
document.write( "cos(angle PAQ) = 0.796162<BR>\n" );
document.write( "angle PAQ = 37.234852<BR>\n" );
document.write( "There appears to be some slight rounding error going on. <BR>\n" );
document.write( "Compare this value to the previous angle PAQ measure found in the section above.<BR>\n" );
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