document.write( "Question 476243: prove that 2arccos(4/5)=arcsin 24/25 \n" ); document.write( "

Algebra.Com's Answer #326555 by ccs2011(207) $\"\"$ $\"About$

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arccos(4/5) equals the angle which satisfies the equation: $\"cos%28phi%29+=+4%2F5\"$
\n" ); document.write( "arcsin(24/25) equals the angle which satisfies the equation: $\"sin%28theta%29+=+24%2F25\"$
\n" ); document.write( "So we are proving that $\"2phi+=+theta\"$
\n" ); document.write( "Lets use a trig property for double angles.
\n" ); document.write( " $\"cos%282phi%29+=+2%2Acos%5E2%28phi%29+-+1\"$
\n" ); document.write( "Substitute $\"cos%28phi%29+=+4%2F5\"$
\n" ); document.write( " $\"cos%282phi%29+=+2%2A%284%2F5%29%5E2+-+1\"$
\n" ); document.write( " $\"cos%282phi%29+=+2%2A%2816%2F25%29+-+1\"$
\n" ); document.write( " $\"cos%282phi%29+=+32%2F25+-+1\"$
\n" ); document.write( " $\"cos%282phi%29+=+32%2F25+-+25%2F25+=+7%2F25\"$
\n" ); document.write( "Now assume $\"2phi+=+theta\"$ is true and check if it works.
\n" ); document.write( " $\"cos%282phi%29+=+cos%28theta%29+=+7%2F25\"$
\n" ); document.write( "Now use a trig identity:
\n" ); document.write( " $\"sin%5E2%28theta%29+%2B+cos%5E2%28theta%29+=+1\"$
\n" ); document.write( " $\"%2824%2F25%29%5E2+%2B+%287%2F25%29%5E2+=+1\"$
\n" ); document.write( " $\"%2824%5E2+%2B+7%5E2%29%2F25%5E2+=+1\"$
\n" ); document.write( " $\"625%2F625+=+1\"$
\n" ); document.write( "It works
\n" ); document.write( "Therefore $\"2phi\"$ is equivalent to $\"theta\"$ . \n" ); document.write( "

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