Which is the value of the sum cos177+cos176+...cos4+cos3 ?

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We'll apply the formula

cos(180-k)=-cos k

-cos 3 = cos (180-3)=cos 177

-cos 4 = cos (180-4)=cos 176

.......................................................

We'll re-write the sum:

S = - cos 3 - cos 4 -.....- cos 89 + cos 90 + cos 89 + ....+cos4 + cos3

We'll eliminate like terms:

**S = 0**

S = cos177 + cos176 + .....+ cos4 + cos3

Let us rewrite:

S = cos(180-3) + cos(180 -4) + ----+ cos4 + cos3

But, we know that :

cos(180 -a) = - cosa

==> S = -cos3 - cos4 + ...+ cos4 + cos3

By eliminating similar,

==> S = 0

Cos177+cos176+ cos 175 + .. cos91+cos90+co89+.. ...cos4 +cos3 . To find the sum.

We know that cos (90+n) = -cos(90-n),

So the given series is equal to Sum of {cos(90+n)+cos(90-n) + cos 90 for n = 1 ,2,3,....97.

= Sum of ( 0) +cos90 , for each n = 1.2,3...87.

= 0. as cos 90 = 0.

cos177+cos176+...+cos4+cos3

=cos(180-3)+cos(180-4)+....+cos4+cos3

=(-cos3)+(-cos4)+.....+cos4+cos3

=0

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