SOLUTION: Find the sum of 1sin(1 degree)+2sin(2 degrees)+3sin(3 degrees)+4sin(4 degrees)+...+178sin(178 degrees)+179sin(179 degrees)+180sin(180 degrees).

Question 1129915: Find the sum of 1sin(1 degree)+2sin(2 degrees)+3sin(3 degrees)+4sin(4 degrees)+...+178sin(178 degrees)+179sin(179 degrees)+180sin(180 degrees).
Answer by ikleyn(52772) (Show Source): You can put this solution on YOUR website!
.

Let  R = 1*sin(1�) + 2*sin(2�) + 3*sin(3�) + . . . + 180*sin(180�).


It is even more convenient for me to write


    R = 0*sin(0�) + 1*sin(1�) + 2*sin(2�) + 3*sin(3�) + . . . + 180*sin(180�)  by adding  0*sin(0�) = 0  as the first term.


Then I can write


    R =   0*sin(0�)   + 1*sin(1�)     + 2*sin(2�)    + 3*sin(3�)     + . . . + 180*sin(180�)     (1)

    R = 180*sin(180�) + 179*sin(179�) + 178*sin178�) + 177*sin(177�) + . . . + 0*sin(0�).        (2)


Note that sin(0�) = sin(180�);  sin(1�) = sin(179�);  sin(2�) = sin(178�);  sin(3�) = sin(177�);  and so on.   
Therefore, adding  (1)  and  (2)


   2R = 180*(sin(0�) + sin(1�) + sin(2�) + sin(3�) + . . . + sin(90�)).


So, our sum R is the same as


     R = 90*(sin(0�) + sin(1�) + sin(2�) + sin(3�) + . . . + sin(90�)).


Now, let z = cos(1�) + i*sin(1�)  is this complex number.


Then, according to the de Moivre's formula


          = cos(2�) + i*sin(2�)

          = cos(3�) + i*sin(3�),

              .  .  .  .  .  .

           = cos(90� + i*sin(90�),

and the sum R  is 90 times the imaginary part  of the sum of the geometric progression

          Q = 1 +  +  +  +  + . . . + .


The sum of the geometric progression is  


     =  = now I will work to extract the imaginary part =  = 

  =  . 


So, the imaginary part has the DENOMINATOR  (cos(1�)-1)^2+sin^2(1�) = 2-2*cos(1�) = 2*(1-cos(1�)).


The imaginary part has the NUMERATOR  sin(91�)*(cos(1�)-1) - (cos(91�)-1)*sin(1�) = sin(91�)*cos(1�) - cos(91�)*sin(1�) - sin(91�) + sin(1�) = 

                                    = sin(90�) - sin(91�) + sin(1�) = 1 - cos(1�) + sin(1�).


Therefore, the answer is:  

    the sum  R = 1*sin(1�) + 2*sin(2�) + 3*sin(3�) + . . . + 180*sin(180�) = .


You can transform it further as you want and/or as you need.


You can even get the numerical value, using your calculator.

Solved.

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