Question

Evaluate without using trigonometric tables :
sin 1 sin 2 sin 3....sin 89

Answer 1

Pair products of sines whose degrees add up to 90 i.e. Sin(1)*Sin(2)*Sin(3)*...*Sin(89)=sin(1)*sin(89) * sin(2)*sin(88) * .... * sin(44)*sin(46) *sin(45) Use the trigonometric product to sum formula, and cos(90)=0 to obtain 2^(-44) * cos(88)*cos(86)*cos(84)* ... *cos(2) * sin(45) = 2^(-44.5) * cos(88)*cos(2) * cos(86)* cos(4) * ... cos(46)*cos(44) I guess that successive application of the product to sum forumula can now be used, but am too lazy to investigate further. Numerical calculation gives an answer of about ~ 2^(-85.75) if it's any help.