document.write( "Question 376518: 2 cos squared (75degrees)-1
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Algebra.Com's Answer #267871 by jsmallt9(3759) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2%2Acos%5E2%2875%29-1\"$
\n" ); document.write( "There are three variations to the formula for cos(2x):

$\"cos%282x%29+=+cos%5E2%28x%29+-+sin%5E2%28x%29\"$
$\"cos%282x%29+=+2%2Acos%5E2%28x%29+-+1\"$
$\"cos%282x%29+=+1+-+2sin%5E2%28x%29\"$

\n" ); document.write( "Your expression fits the middle variation with x = 75. Using the pattern for this formula, with 75 in the place of x, we get:
\n" ); document.write( "cos(2*75)
\n" ); document.write( "which simplifies to
\n" ); document.write( "cos(150)
\n" ); document.write( "The reference angle for 150, since it is in the second quadrant, is 180 - 150 or 30. This is one of the special angles whose sin's and cos's you are supposed to learn. cos(30) = $\"sqrt%283%2F2%29\"$ And since 150 is in the second quadrant and cos is negative in the second quadrant...
\n" ); document.write( "cos(150) = $\"-sqrt%283%29%2F2\"$

\n" ); document.write( "Note: If you simply use your calculator on the original expression, you will end up with a decimal. It would be difficult to recognize this decimal as $\"sqrt%283%29%2F2\"$ . The decimal is an approximate answer. The square root answer is 100% exactly correct. The moral of the story, if you can use special angles on a problem, do so. You will end up with exact answers instead of decimal approximations. This is why we used the cos(2x) formula. It turned and expression with a non-special angle, 75, into an expression involving a special angle, 150. \n" ); document.write( "

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