Question

Sin 1+sin 2+sin3............sin89

Answer 1

Observe that sinx * sin(60 - x) * sin(60 + x) = 1/4 * sin(3x), as:

2 sinx sin(60-x) = cos(x-60+x) - cos(x+60-x) = cos(2x-60) - cos60 = cos(2x-60) - 1/2

Therefore:

4 sinx * sin(60 - x) * sin(60 + x) = 2 [ 2 sinx sin(60-x) ] sin(60+x) =

= 2 [ cos(2x-60) - 1/2 ] sin(60+x) = 2 cos(2x-60) sin(60+x) - 2 1/2 sin(60+x) =

= sin(60+x+2x-60) + sin(60+x-2x+60) - sin(60+x) =

= sin(3x) + sin(120-x) - sin(60+x) = sin(3x)

[since 120 - x + 60 + x = 180, sin(120-x) = sin(60+x)]

Therefore: sinx * sin(60 - x) * sin(60 + x) = 1/4 * sin(3x)

So:

sin1 * sin2 * sin3 * .......... * sin89 =

[ sin1 sin59 sin61 ] * [ sin2 sin58 sin62] * [sin3 sin57 sin63] * ...... *

* [sin29 sin31 sin89] * sin30 * sin60 = (29 triplets there)

= 1/4 sin3 * 1/4 sin6 * 1/4 sin9 * ....* 1/4 sin87 * sin30 * sin60 =

= (1/4)^29 sin3 sin6 .... sin87 * 1/2 * sqrt(3)/2 =

= sqrt(3) * (1/4)^30 * sin3 sin6 .... sin87 =

= sqrt(3) * (1/4)^30 * [sin3 sin57 sin63] * [sin6 sin54 sin66] * [sin9 sin 51 sin69] * .....

* [sin27 sin33 sin87] * sin30 * sin60 = (by similar argument - 9 triplets now)

= sqrt(3) * (1/4)^30 * (1/4)^9 * sin9 sin18 sin 27 .... sin81 * 1/2 * sqrt(3)/2 =

= 3 * (1/4)^40 * sin9 sin18 sin27 .... sin 81 =

= 3 * (1/4)^40 * [sin9 sin81] [sin18 sin72] [sin27 sin63] [sin36 sin54] [sin45] =

= 3 * (1/4)^40 * sqrt(2)/2 * (1/2)^4 * [2 sin9 sin81] [2 sin18 sin72] [2 sin27 sin63] [2 sin36 sin54] =

= 3 * sqrt(2)/2 * (1/4)^42 * [2 sin9 cos9] [2 sin18 cos18] [2 sin27 cos27] [2 sin36 cos36] =

= 3 * sqrt(2)/2 * (1/4)^42 * sin18 sin36 sin54 sin72 =

= 3 * sqrt(2)/2 * (1/4)^42 * (1/2)^2 * [2 sin18 sin72] [2 sin36 sin54] =

= 3 * sqrt(2)/2 * (1/2)^84 * (1/2)^2 * [2 sin18 cos18] [2 sin36 cos36] =

= 3 * sqrt(2)/2 * (1/2)^84 * (1/2)^2 * sin36 sin72 =

[Note: sin36 and sin72 are easy to calculate iteratively, but you can consult wolframalpha which yields closed formulas for both]

= 3 * sqrt(2) * (1/2)^87 * [sqrt(5/8-sqrt(5)/8)] [sqrt(5/8+sqrt(5)/8)] =

= 3 * sqrt(2) * (1/2)^87 * sqrt[ (5/8)^2 - 5/8^2 ] =

= 3 * sqrt(2) * (1/2)^87 * sqrt[ 25/8^2 - 5/8^2 ] =

= 3 * sqrt(2) * (1/2)^87 * sqrt(20) / 8 = 3 * sqrt(2) * (1/2)^87 * 2 sqrt(5) / 8 =

= 3 * sqrt(2) * (1/2)^87 * sqrt(5) / 4 = 3 * sqrt(2) * (1/2)^89 * sqrt(5) =

= 3 sqrt(10) / 2^89

which is the exact value for sin1 sin2 sin3 .... sin89