document.write( "Question 1210192: Find an acute angle A such that \sin 4A = \sin A + \sin 2A. Express your answer in degrees. \n" ); document.write( "
Algebra.Com's Answer #851528 by ikleyn(52858)\"\" \"About 
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\n" ); document.write( "Find an acute angle A such that sin 4A = sin A + sin 2A. Express your answer in degrees.
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\n" ); document.write( "\n" ); document.write( "        In this my post,  I will give a standard solution to this problem.\r
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\n" ); document.write( "\n" ); document.write( "        Saying \" a standard solution \",  I mean a tradition method,  by which I \r
\n" ); document.write( "\n" ); document.write( "        (and thousands and millions other students of my generation)  solved such problems in a class,
\n" ); document.write( "        considering it as a routine.\r
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document.write( "Start from\r\n" );
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document.write( "    sin(4A) = sin(A) + sin(2A).    (1)\r\n" );
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document.write( "Transform into\r\n" );
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document.write( "    sin(4A) - sin(2A) = sin(A).\r\n" );
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document.write( "Use the common/general formula of Trigonometry\r\n" );
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document.write( "    sin(a) - sin(b) = \"2cos%28%28a%2Bb%29%2F2%29%2Asin%28%28a-b%29%2F2%29\",\r\n" );
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document.write( "valid for any angles 'a' and 'b'.  Apply it for a = 4A, b = 2A.  \r\n" );
document.write( "You will get\r\n" );
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document.write( "    2cos(3A)*sin(A) = sin(A).    (2)\r\n" );
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document.write( "One solution is sin(A) = 0,  which implies  A = 0 or \"pi\", \r\n" );
document.write( "which in degrees is A = 0° or A = 180°.\r\n" );
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document.write( "Next, if we want to find other solutions with sin(A) =/= 0, then \r\n" );
document.write( "we can cancel sin(A) in both sides of equation (2).  You will get then\r\n" );
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document.write( "    2cos(3A) = 1,  or  cos(3A) = \"1%2F2\".\r\n" );
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document.write( "It has the solutions\r\n" );
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document.write( "    3A = \"pi%2F3+%2B+2k%2Api\"  or  3A = \"5pi%2F3%2B2k%2Api\",  k = 0. +/-1, +/-2, . . . \r\n" );
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document.write( "which implies\r\n" );
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document.write( "    A = \"pi%2F9%2B%282%2F3%29%2Ak%2Api\"  or  A = \"5pi%2F9%2B%282%2F3%29%2Ak%2Api\",  k = 0. +/-1, +/-2, . . . \r\n" );
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document.write( "In degrees, it is\r\n" );
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document.write( "    A = 20° + 120°*k  or  A = 100° + 120°*k.\r\n" );
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document.write( "ANSWER.  The solutions are  k*180°,  20° + 120*k,  100° + 120*k,  k = 0, +/-1, +/-2, . . . \r\n" );
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document.write( "         If you want to list the angles in the interval [0°,360°), they are\r\n" );
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document.write( "         0°, 20°, 100°, 140°, 220°, 260°, 340°.\r\n" );
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document.write( "         An acute angles-solutions are 0° (degenerated) and 20°.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus I got all the solutions, without missing no one.\r
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\n" ); document.write( "\n" ); document.write( "I prepared a plot of both sides of the original equation (1) in the interval from \"-2pi\" to \"2pi\".
\n" ); document.write( "See the link\r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/cgohxl603j \r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/cgohxl603j\r
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\n" ); document.write( "\n" ); document.write( "Looking at this plot, you may see all the roots as the intersections of the plotted curves.\r
\n" ); document.write( "\n" ); document.write( "You may count the number of the roots in the interval [0°,360°),
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