document.write( "Question 629284: If sin[sin^-1( $\"1%2F5\"$ ) + cos^-1(x)] = 1, then x = ?
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Algebra.Com's Answer #396198 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "sin[sin^-1() + cos^-1(x)] = 1\r\n" );
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document.write( "Let a = sin^-1() and b = cos^-1(x)\r\n" );
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document.write( "Then sin(a) =  and cos(b) = x\r\n" );
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document.write( "And now our equation is \r\n" );
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document.write( "sin(a + b) = 1\r\n" );
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document.write( "Using an identity on the left side,\r\n" );
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document.write( "sin(a)cos(b) + cos(a)sin(b) = 1\r\n" );
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document.write( "Draw two right triangles, one with acute angle a and one with acute angle b:\r\n" );
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document.write( "Since the sine of a is  and since , put\r\n" );
document.write( "the numerator of , which is 1, on the side opposite a and \r\n" );
document.write( "the denominator of  which is 5, on the hypotenuse.\r\n" );
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document.write( "Since the cosine of b is x, we consider that as the fraction \r\n" );
document.write( "and since , put the numerator of , which is x,\r\n" );
document.write( "on the side adjacent b and the denominator of  which is 1, on the\r\n" );
document.write( "hypotenuse.\r\n" );
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document.write( "Next we use the Pythagorean theorem to find the side adjacent a, and\r\n" );
document.write( "the side opposite b:\r\n" );
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document.write( "a² + b² = c²              a² + b² = c²\r\n" );
document.write( "a² + 1² = 5²              x² + b² = 1²\r\n" );
document.write( " a² + 1 = 25              x² + b² = 1\r\n" );
document.write( "     a² = 24                   b² = 1 - x² \r\n" );
document.write( "      a =                    b =              \r\n" );
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document.write( "Now we can use the right triangles above and their sides to substitute\r\n" );
document.write( "into the equation:\r\n" );
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document.write( "sin(a)cos(b) + cos(a)sin(b) = 1\r\n" );
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document.write( "()() + ()() = 1\r\n" );
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document.write( "Clear of fractions by multiplying through by LCD 5\r\n" );
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document.write( "x +  = 5\r\n" );
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document.write( "Isolate the radical term:\r\n" );
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document.write( "     = 5 - x\r\n" );
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document.write( "Square both sides of the equations:\r\n" );
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document.write( "       24(1 - x²) = (5 - x)²\r\n" );
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document.write( "        24 - 24x² = (5 - x)(5 - x)\r\n" );
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document.write( "        24 - 24x² = 25 - 10x + x²\r\n" );
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document.write( "Get 0 on the right:\r\n" );
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document.write( "   25x² - 10x + 1 = 0\r\n" );
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document.write( "Factor the left side:\r\n" );
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document.write( " (5x - 1)(5x - 1) = 0\r\n" );
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document.write( "        (5x - 1)² = 0 \r\n" );
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document.write( "           5x - 1 = 0\r\n" );
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document.write( "               5x = 1\r\n" );
document.write( "                x =     \r\n" );
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document.write( "Edwin
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