document.write( "Question 881250: If sec(a+15)=csc(2a) , find the smallest positive value of a, in degrees?
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Algebra.Com's Answer #532229 by Edwin McCravy(20065)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"matrix%281%2C3%2C++sec%28a%2B%2215%B0%22%29%2C+%22%22=%22%22%2C+csc%282a%29++%29\"\r\n" );
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document.write( "\"matrix%281%2C3%2C+1%2Fcos%28a%2B%2215%B0%22%29%2C+%22%22=%22%22%2C+1%2Fsin%282a%29++%29\"\r\n" );
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document.write( "Cross-multiply\r\n" );
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document.write( "\"matrix%281%2C3%2C++sin%282a%29%2C+%22%22=%22%22%2C+cos%28a%2B%2215%B0%22%29++%29\"\r\n" );
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document.write( "We need to get that difference to a product.  One well-known\r\n" );
document.write( "difference to product formula is\r\n" );
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document.write( "We can use it if we can write sin(2a) as a cosine.  We can\r\n" );
document.write( "do that by using the cofunction identity, and write sin(2a) = cos(90°-2a)\r\n" );
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document.write( "Use the identity with x = 90°-2a and y = a+15°\r\n" );
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document.write( " \"%22%22=%22%22\" \"%220%22\"\r\n" );
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document.write( " \"%22%22=%22%22\" \"%220%22\"\r\n" );
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document.write( "\"-2sin%28%28%22105%B0%22-a%29%2F2%29sin%28%28%2275%B0%22-3a%29%2F2%29+%29\" \"%22%22=%22%22\" \"%220%22\"\r\n" );
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document.write( "Divide both sides by -2\r\n" );
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document.write( "\"sin%28%28%22105%B0%22-a%29%2F2%29sin%28%28%2275%B0%22-3a%29%2F2%29+%29\" \"%22%22=%22%22\" \"%220%22\"\r\n" );
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document.write( "Use the zero-factor property:\r\n" );
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document.write( "\"sin%28%28%22105%B0%22-a%29%2F2%29+%29\" \"%22%22=%22%22\" \"%220%22\"    and      \"sin%28%28%2275%B0%22-3a%29%2F2%29+%29\" \"%22%22=%22%22\" \"%220%22\"\r\n" );
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document.write( "\"%28%22105%B0%22-a%29%2F2%29+%29\" \"%22%22=%22%22\" \"%22180%B0%22n\"    and      \"%28%2275%B0%22-3a%29%2F2\"\"%22%22=%22%22\" \"%22180%B0%22n\"\r\n" );
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document.write( "Multiply both equations through by 2\r\n" );
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document.write( "105°-a = 360°n             and       75²-3a = 360°n\r\n" );
document.write( "      -a = 360°n-105°                   -3a = 360°n-75°\r\n" );
document.write( "       a = -360°n+105°                    a = -120°n+25°\r\n" );
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document.write( "Both have their smallest positive value when n=0\r\n" );
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document.write( "       a = -360°(0)+105°                  a = -120°(0)+25°\r\n" );
document.write( "       a = 105°                           a = 25°\r\n" );
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document.write( "So the smallest positive value of a is 25°\r\n" );
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document.write( "Edwin
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