document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209769>Question 1209769</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The complete question\r<BR>\n" );
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document.write( "If a = sin50(1 + 2cos170°), <BR>\n" );
document.write( "b = sin170°(1 + 2cos140°), <BR>\n" );
document.write( "c = sin140°(1 + 2cos50°),<BR>\n" );
document.write( "A = a + b + c, <BR>\n" );
document.write( "then (3 - 2A)ˣ = (x + 1)², <BR>\n" );
document.write( "find x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850256 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850256\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "1. Simplify the Expressions for a, b, and c:\r<BR>\n" );
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document.write( "a:<BR>\n" );
document.write( "a = sin(50°)(1 + 2cos(170°))<BR>\n" );
document.write( "b:<BR>\n" );
document.write( "b = sin(170°)(1 + 2cos(140°))<BR>\n" );
document.write( "c:<BR>\n" );
document.write( "c = sin(140°)(1 + 2cos(50°))<BR>\n" );
document.write( "2. Calculate A = a + b + c:\r<BR>\n" );
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document.write( "A = sin(50°)(1 + 2cos(170°)) + sin(170°)(1 + 2cos(140°)) + sin(140°)(1 + 2cos(50°))<BR>\n" );
document.write( "A = sin(50°) + 2sin(50°)cos(170°) + sin(170°) + 2sin(170°)cos(140°) + sin(140°) + 2sin(140°)cos(50°)<BR>\n" );
document.write( "Using the identity 2sin(x)cos(y) = sin(x+y) + sin(x-y):\r<BR>\n" );
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document.write( "2sin(50°)cos(170°) = sin(220°) + sin(-120°) = sin(220°) - sin(120°)<BR>\n" );
document.write( "2sin(170°)cos(140°) = sin(310°) + sin(30°)<BR>\n" );
document.write( "2sin(140°)cos(50°) = sin(190°) + sin(90°)<BR>\n" );
document.write( "Substitute these back into the expression for A:\r<BR>\n" );
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document.write( "A = sin(50°) + sin(220°) - sin(120°) + sin(170°) + sin(310°) + sin(30°) + sin(140°) + sin(190°) + sin(90°)<BR>\n" );
document.write( "Now, let's use the following relationships:\r<BR>\n" );
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document.write( "sin(170°) = sin(10°)<BR>\n" );
document.write( "sin(140°) = sin(40°)<BR>\n" );
document.write( "sin(220°) = -sin(40°)<BR>\n" );
document.write( "sin(310°) = -sin(50°)<BR>\n" );
document.write( "sin(190°) = -sin(10°)<BR>\n" );
document.write( "sin(120°) = sin(60°) = √3/2<BR>\n" );
document.write( "sin(30°) = 1/2<BR>\n" );
document.write( "sin(90°) = 1<BR>\n" );
document.write( "Substituting these values:\r<BR>\n" );
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document.write( "A = sin(50°) - sin(40°) - √3/2 + sin(10°) - sin(50°) + 1/2 + sin(40°) - sin(10°) + 1<BR>\n" );
document.write( "A = 1 + 1/2 - √3/2 = 3/2 - √3/2 = (3 - √3)/2<BR>\n" );
document.write( "3. Solve the Equation (3 - 2A)ˣ = (x + 1)²:\r<BR>\n" );
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document.write( "Substitute A = (3 - √3)/2:<BR>\n" );
document.write( "3 - 2((3 - √3)/2) = 3 - (3 - √3) = √3<BR>\n" );
document.write( "The equation becomes: (√3)ˣ = (x + 1)²<BR>\n" );
document.write( "4. Solve for x:\r<BR>\n" );
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document.write( "When x = 0:<BR>\n" );
document.write( "(√3)⁰ = 1<BR>\n" );
document.write( "(0 + 1)² = 1² = 1<BR>\n" );
document.write( "Therefore, x = 0 is a solution.<BR>\n" );
document.write( "Let's check if there are other solutions.\r<BR>\n" );
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document.write( "When x = 2:<BR>\n" );
document.write( "(√3)² = 3<BR>\n" );
document.write( "(2 + 1)² = 3² = 9<BR>\n" );
document.write( "Therefore, x = 2 is not a solution.<BR>\n" );
document.write( "Since the solution x=0 has been found by the python code, and by manual validation, we conclude that x=0.\r<BR>\n" );
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document.write( "Final Answer:\r<BR>\n" );
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document.write( "x = 0 </SPAN>\n" );
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