Question

Sin(1) + sin (2) + sin (3) + .........sin(89)

Answer 1

Hi friend,


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 the lazy way is to 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


Hope this helps you and not confuses you...

If you have any dout please ask me....

Please mark it as brainliest answer...☺☺☺