SOLUTION: find the exact value by using a half-angle identity. sin 22.5degrees show your work. thanks!

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Question 237746: find the exact value by using a half-angle identity.
sin 22.5degrees
show your work. thanks!
Answer by jsmallt9(3758) About Me  (Show Source):
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.

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:
sin%28%281%2F2%29x%29+=+0+%2B-+sqrt%28%281-cos%28x%29%29%2F2%29
Substituting 45 in for x:
sin%28%281%2F2%2945%29+=+0+%2B-+sqrt%28%281-cos%2845%29%29%2F2%29
sin%2822.5%29+=+0+%2B-+sqrt%28%281-+sqrt%282%29%2F2%29%2F2%29
Since 22.5 is in the first quadrant and sin is positive in the first quadrant, we will use the positive square root:
sin%2822.5%29+=+sqrt%28%281-+sqrt%282%29%2F2%29%2F2%29
The only thing left to do is simplify the expression within the outer square root:
sin%2822.5%29+=+sqrt%28%282%2F2-+sqrt%282%29%2F2%29%2F2%29
sin%2822.5%29+=+sqrt%28%28%282+-+sqrt%282%29%29%2F2%29%2F2%29
sin%2822.5%29+=+sqrt%28%28%28%282+-+sqrt%282%29%29%2F2%29%2F2%29%282%2F2%29%29

sin%2822.5%29+=+sqrt%28%282+-+sqrt%282%29%29%2F4%29
sin%2822.5%29+=+sqrt%282+-+sqrt%282%29%29%2Fsqrt%284%29
sin%2822.5%29+=+sqrt%282+-+sqrt%282%29%29%2F2