document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206316>Question 1206316</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you solve sin x + square root of 2 = - sin x on the interval -pi/2 < x < pi/2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843673 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=843673\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We can use a spreadsheet to check each answer mentioned (x = -pi/2 and x = -pi/4)<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >LHS</TD><TD >RHS</TD><TD >LHS = RHS?</TD></tr><tr><TD >-pi/2</TD><TD >0.414214</TD><TD >1</TD><TD >No</TD></tr><tr><TD >-pi/4</TD><TD >0.707107</TD><TD >0.707107</TD><TD >Yes</TD></tr>\n" );
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document.write( "The table shows that only x = -pi/4 works.<BR>\n" );
document.write( "LHS = left hand side<BR>\n" );
document.write( "RHS = right hand side\r<BR>\n" );
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document.write( "Furthermore,<BR>\n" );
document.write( "-pi/2 < -pi/4 < pi/2<BR>\n" );
document.write( "Multiply all sides by -4<BR>\n" );
document.write( "2pi > pi > -2pi<BR>\n" );
document.write( "which flips to<BR>\n" );
document.write( "-2pi < pi < 2pi\r<BR>\n" );
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document.write( "The last inequality <BR>\n" );
document.write( "-2pi < pi < 2pi<BR>\n" );
document.write( "being true leads back to <BR>\n" );
document.write( "-pi/2 < -pi/4 < pi/2<BR>\n" );
document.write( "also being true\r<BR>\n" );
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document.write( "Or you could use these decimal approximations<BR>\n" );
document.write( "pi/2 = 1.570796<BR>\n" );
document.write( "pi/4 = 0.785398<BR>\n" );
document.write( "So<BR>\n" );
document.write( "-pi/2 < -pi/4 < pi/2<BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "-1.570796 < -0.785398 < 1.570796\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Another way to verify the answer is to graph h(x) = f(x) - g(x)<BR>\n" );
document.write( "where<BR>\n" );
document.write( "f(x) = sin(x) + sqrt(2)<BR>\n" );
document.write( "g(x) = -sin(x)\r<BR>\n" );
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document.write( "Or basically you need to graph h(x) = 2*sin(x)+sqrt(2)<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/s6agjxpf7d\">https://www.desmos.com/calculator/s6agjxpf7d</a><BR>\n" );
document.write( "Desmos is free graphing software. GeoGebra is also another good choice.<BR>\n" );
document.write( "The curve intersects the x axis at (-pi/4, 0) to confirm that x = -pi/4 is the only solution to this equation on this specified domain interval.\r<BR>\n" );
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document.write( "If the -pi/2 < x < pi/2 portion wasn't required, then there would be infinitely many solutions.<BR>\n" );
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