document.write( "Question 237746: find the exact value by using a half-angle identity.
\n" ); document.write( "sin 22.5degrees
\n" ); document.write( "show your work. thanks! \n" ); document.write( "
Algebra.Com's Answer #174798 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Whenever a Trig problem refers to \"exact\" values, it implies that an angle which is a multiple of 30 or 45 is involved since these are the only angles for which exact values can be found.

\n" ); document.write( "So in your problem we need to \"connect\" sin(22.5) with an angle that is a multiple of 30 or 45. This is not difficult since 22.5 = 45/2. So we'll use the half-angle formula formula for sin:
\n" ); document.write( "\"sin%28%281%2F2%29x%29+=+0+%2B-+sqrt%28%281-cos%28x%29%29%2F2%29\"
\n" ); document.write( "Substituting 45 in for x:
\n" ); document.write( "\"sin%28%281%2F2%2945%29+=+0+%2B-+sqrt%28%281-cos%2845%29%29%2F2%29\"
\n" ); document.write( "\"sin%2822.5%29+=+0+%2B-+sqrt%28%281-+sqrt%282%29%2F2%29%2F2%29\"
\n" ); document.write( "Since 22.5 is in the first quadrant and sin is positive in the first quadrant, we will use the positive square root:
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%28%281-+sqrt%282%29%2F2%29%2F2%29\"
\n" ); document.write( "The only thing left to do is simplify the expression within the outer square root:
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%28%282%2F2-+sqrt%282%29%2F2%29%2F2%29\"
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%28%28%282+-+sqrt%282%29%29%2F2%29%2F2%29\"
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%28%28%28%282+-+sqrt%282%29%29%2F2%29%2F2%29%282%2F2%29%29\"
\n" ); document.write( "
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%28%282+-+sqrt%282%29%29%2F4%29\"
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%282+-+sqrt%282%29%29%2Fsqrt%284%29\"
\n" ); document.write( "\"sin%2822.5%29+=+sqrt%282+-+sqrt%282%29%29%2F2\"\r
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