Sine Product
If the value of the expression above can be represented by , where and are coprime positive integers, compute .
The answer is 17.
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sin ( 3 4 π ) sin ( 3 4 9 π ) sin ( 3 4 1 3 π ) sin ( 3 4 1 5 π ) = = = = = = = = cos ( 1 7 π ) cos ( 1 7 2 π ) cos ( 1 7 4 π ) cos ( 1 7 8 π ) 2 sin ( 1 7 π ) 2 sin ( 1 7 π ) cos ( 1 7 π ) cos ( 1 7 2 π ) cos ( 1 7 4 π ) cos ( 1 7 8 π ) 4 sin ( 1 7 π ) 2 sin ( 1 7 2 π ) cos ( 1 7 2 π ) cos ( 1 7 4 π ) cos ( 1 7 8 π ) 8 sin ( 1 7 π ) 2 sin ( 1 7 4 π ) cos ( 1 7 4 π ) cos ( 1 7 8 π ) 1 6 sin ( 1 7 π ) 2 sin ( 1 7 8 π ) cos ( 1 7 8 π ) 1 6 sin ( 1 7 π ) sin ( 1 7 1 6 π ) 1 6 sin ( 1 7 π ) sin ( 1 7 π ) 1 6 1
Thus, p + q = 1 + 1 6 = 1 7
Notes:
sin θ = cos ( 2 π − θ ) = sin ( π − θ )
sin ( 2 θ ) = 2 sin θ cos θ