Explanation:
x/2 = 3×x/6
sin(x/2) = sin(3x/6)
cos(x/2) = cos(3x/6)
tan(x/2) = tan(3x/6) = sin(3x/6)/cos(3x/6)
looking up triple angle identities for trigonometric functions we find :
cos(3x) = 4cos³(x) - 3cos(x)
so,
cos(x/2) =
cos(3x/6) = 4cos³(x/6) - 3cos(x/6) = 4(3/5)³ - 3×3/5 =
= 4×27/125 - 9/5 = 108/125 - 9×25/125 =
= 108/125 - 225/125 = -117/125 =
= -0.936
sin(3x) = 3×sin(x) - 4×sin³(x)
so,
sin(x/2) =
sin(3×x/6) = 3×sin(x/6) - 4×sin³(x/6)
also remember,
sin²(x) + cos²(x) = 1
therefore,
sin²(x/6) + cos²(x/6) = 1
sin(x/6) = sqrt(1 - cos²(x/6)) = sqrt(1 - (3/5)²) =
= sqrt(1 - 9/25) = sqrt(25/25 - 9/25) =
= sqrt(16/25) = 4/5 = 0.8
so, again
sin(x/2) =
sin(3×x/6) = 3×sin(x/6) - 4×sin³(x/6) =
= 3×4/5 - 4×(4/5)³ =
= 12/5 - 4×64/125 = 12×25/125 - 256/125 =
= 300/125 - 256/125 = 44/125 =
= 3×0.8 - 4×0.8³ = 0.352
tan(x/2) = sin(x/2)/cos(x/2) =
= 44/125 / -117/125 = -44/117 =
= 0.352 / -0.936 =
= -0.376068376...