document.write( "Question 1209808: Find the value of θ,
\n" ); document.write( "sin20°sinθ + sin100°sin(20 - θ)° = 0 \n" ); document.write( "

Algebra.Com's Answer #851871 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Find the value of θ,
\n" ); document.write( "sin20°sinθ + sin100°sin(20 - θ)° = 0
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\n" ); document.write( "\n" ); document.write( " In the post by @CPhill, his solution, giving the answer $\"theta\"$ = 40°, is INCORRECT.\r
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document.write( "Let's check it.\r\n" );
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document.write( "We have   = sin(20°)*sin(40°) = 0.34202014332*0.64278760968 = 0.21984631.\r\n" );
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document.write( "Next, we have  \r\n" );
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document.write( "     = cos(10°)*sin(20°-40°) = cos(10°)*sin(-20°) = 0.98480775301*(-0.34202014332) = -0.336824089.\r\n" );
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document.write( "Thus \r\n" );
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document.write( "     +  = sin(20°)*sin(40°) + cos(10°)*sin(-20°) = 0.21984631 + (-0.336824089) = -0.116977779.\r\n" );
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document.write( "is not zero.\r\n" );
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\n" ); document.write( "\n" ); document.write( " Thus the answer by @CPhill is disproved.\r
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\n" ); document.write( "\n" ); document.write( "The right solution can be found using numerical methods.\r
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\n" ); document.write( "\n" ); document.write( "I used a plotting tool in web-site https:\\www.desmos.com/calculator/\r
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\n" ); document.write( "\n" ); document.write( "It produces plots and is smart enough to make all necessary accompanying calculations automatically.\r
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\n" ); document.write( "\n" ); document.write( "See my plot of participating functions in this web-page\r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/da6cvuhjij\r
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\n" ); document.write( "\n" ); document.write( "https://www.desmos.com/calculator/da6cvuhjij\r
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\n" ); document.write( "\n" ); document.write( "Our solutions are the intersection points of the plots.\r
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\n" ); document.write( "\n" ); document.write( "One intersection point is x= 0.5236 radians, or 30 degrees.\r
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\n" ); document.write( "\n" ); document.write( "Another intersection point is x = 3.66519 radians, or 210 degrees.\r
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\n" ); document.write( "\n" ); document.write( "// To see the coordinates of the intersection points, click on these points.\r
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\n" ); document.write( "\n" ); document.write( "So, numerically we get this answer: the angle $\"theta\"$ may have two values : 30° and/or 210°.\r
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\n" ); document.write( "\n" ); document.write( " Now I will $\"highlight%28highlight%28PROVE%29%29\"$ to you $\"highlight%28highlight%28MATHEMATICALLY%29%29\"$ that these answers are correct.\r
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document.write( "Indeed, for   = 30°\r\n" );
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document.write( "     = sin(20°)*sin(30°) =    <<<---===  since  sin(30°) = .\r\n" );
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document.write( "     = cos(10°)*sin(-10°) =  =   <<<---=== sinse  sin(a)*cos(a) = .\r\n" );
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document.write( "Now you see that  = 30° is the solution: it is PROVED.\r\n" );
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document.write( "Similar proof works for  = 210°.\r\n" );
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\n" ); document.write( "\n" ); document.write( "At this point, the wrong solution of @CPhill is disproved completely,
\n" ); document.write( "and the right solution is found numerically and proved mathematically.\r
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