# Calculate the sum s=cos1+cos2+cos3+...+cos179

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this problem is very easy ....for beginning ......you have to write the sum: S=cos1+cos2+.......+cos179

and now you have to write this sum in reverse way :

like that:

S=cos179+cos178+........................+cos1

S=cos1 +cos2 +.........................+cos179( you have to gather)

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2S=(cos179+cos1)+(cos178+cos2)+................+(cos1+cos179)

and now you can see that:

cos 179+cos1=2x((cos179+1):2) x(cos(179-1):2)=0

cos 178+cos2=0

2S=0 S=0

Using the formula cos(180-k)=-cos k

-cos 179 = cos (180-179)=cos 1

-cos 178 = cos (180-178)=cos 2 and so on.

So the sum could be written in this way:

-cos 179 -cos 178 -cos 177 - ... +cos 90 +...+cos 177 + cos 178 + cos 179.

Reducing the similar terms, we'll have s=0