what would be the answer to this problem?
cos 45 = cos ( 2 * 22.5 ) = 1 - 2 sin^2 ( 22.5 )
1/sq rt ( 2 ) = 1 - 2 sin^2 ( 22. 5 )
1 = sq rt ( 2 ) - 2 sq rt ( 2 ) sin^2 ( 22. 5 )
2 sq rt ( 2 ) sin^2 ( 22.5 ) = sq rt ( 2 ) - 1
sin^2 ( 22.5 ) = [ sq rt ( 2 ) - 1 ] / 2 sq rt ( 2 )
sin^2 ( 22.5 ) = sq rt ( 2 ) [ sq rt ( 2 ) - 1 ] / 4
sin^2 ( 22.5 ) = [ 2 - sq rt ( 2 ) ]/4
sin 22.5 = +/- sq rt ( 2 - sq rt ( 2 ) ) / 2
sin(22.5°) = sin(45°/2) = sqrt(1-cos(45°))/sqrt(2)
=sqrt(1-1/sqrt(2))/sqrt(2)
=sqrt(2-sqrt(2))/2
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Verified answer
cos 45 = cos ( 2 * 22.5 ) = 1 - 2 sin^2 ( 22.5 )
1/sq rt ( 2 ) = 1 - 2 sin^2 ( 22. 5 )
1 = sq rt ( 2 ) - 2 sq rt ( 2 ) sin^2 ( 22. 5 )
2 sq rt ( 2 ) sin^2 ( 22.5 ) = sq rt ( 2 ) - 1
sin^2 ( 22.5 ) = [ sq rt ( 2 ) - 1 ] / 2 sq rt ( 2 )
sin^2 ( 22.5 ) = sq rt ( 2 ) [ sq rt ( 2 ) - 1 ] / 4
sin^2 ( 22.5 ) = [ 2 - sq rt ( 2 ) ]/4
sin 22.5 = +/- sq rt ( 2 - sq rt ( 2 ) ) / 2
sin(22.5°) = sin(45°/2) = sqrt(1-cos(45°))/sqrt(2)
=sqrt(1-1/sqrt(2))/sqrt(2)
=sqrt(2-sqrt(2))/2