document.write( "Question 942251: cos(2θ+20)=sin(3θ−10)\r
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Algebra.Com's Answer #854628 by ikleyn(53937) $\"\"$ $\"About$

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\n" ); document.write( "cos(2x+20°) = sin(3x−10°). Find x.
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\n" ); document.write( "\n" ); document.write( "        The solution in the post by @lwsshar3 is a part of the truth, but not the whole truth.\r
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\n" ); document.write( "\n" ); document.write( "        The solution by @lwsshar3 is correct if and only if both angles, 2x+20° and 3x-10° are in QI.\r
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\n" ); document.write( "\n" ); document.write( "        But if we look for all possible angles x between 0° and 360° (360° is the period for both functions\r
\n" ); document.write( "\n" ); document.write( "        in the left and the right side of the equation, so it is NATURAL to investigate this interval), \r
\n" ); document.write( "\n" ); document.write( "        then his solution is INCOMPLETE, because there are 5 (five) different angles 'x' satisfying the given equation \r
\n" ); document.write( "\n" ); document.write( "        in interval [0°,360°).\r
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\n" ); document.write( "\n" ); document.write( "        And this is the whole truth. See my solution below.\r
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document.write( "Equation\r\n" );
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document.write( "    cos(2x+20°) = sin(3x−10°)    (1)\r\n" );
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document.write( "implies\r\n" );
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document.write( "    (2x+20°) + (3x-10°) = = 90° + k*360°, k = 0, +/-1, +/-2, . . . (2)\r\n" );
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document.write( "because both functions tan and cot are periodical with period .\r\n" );
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document.write( "So, we can rewrite equation (2) in the form\r\n" );
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document.write( "    5x = 80° + k*360°, k = 0, 1, 2, 3, 4.\r\n" );
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document.write( "We can limit ourselves with 5 values of k, because for other values of 'k', the angles 'x' will be geometrically the same,\r\n" );
document.write( "but formally will be outside of the interval [0°,360°).\r\n" );
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document.write( "For these 5 values of 'k', we will have 5 different values of x in the interval [0°,360°)\r\n" );
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document.write( "    x = 16°,  16°+72° = 88°,  16°+144° = 160°,  16°+216° = 232°,  16°+288° = 304°.\r\n" );
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document.write( "ANSWER. Taking into account the periodicity of the left side and the right side of the given equation, \r\n" );
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document.write( "it is natural to look for solutions 'x' in the interval [0°,360°).\r\n" );
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document.write( "In this interval, the given equation has 5 (five) different solutions 16°, 88°, 160°, 232° and 302°.\r\n" );
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\n" ); document.write( "\n" ); document.write( "//////////// I N T E R E S T I N G ////////////\r
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\n" ); document.write( "\n" ); document.write( "Driven by my curiosity, I posted today (May 21, 2026, about 3:30 PM) this problem to Google Overview Artificial Intelligence
\n" ); document.write( "to look how it treats the problem.\r
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\n" ); document.write( "\n" ); document.write( "It treats it precisely as in the post by @lwsshar3 - so, its analysis is incomplete.\r
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\n" ); document.write( "\n" ); document.write( "Does it embarrass me? - - - Both \"Yes\" and \"No\".\r
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\n" ); document.write( "\n" ); document.write( "It simply demonstrates the level and the style of work of contemporary AI.\r
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\n" ); document.write( "\n" ); document.write( "It can not think as a human can (because it is NOT ABLE to think as a human can),
\n" ); document.write( "but simply re-writes from the storage of existing solutions, with no thinking.\r
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