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Find the value of $\cos 1^\circ + \cos 2^\circ + \cos 3^\circ +.....+\cos 180^\circ $

My Attempt: $$=\cos 1^\circ + \cos 2^\circ + \cos 3^\circ +....+\cos 180^\circ $$ $$= \cos 1^\circ + \cos 2^\circ + \cos 3^\circ +....-1$$

How do I complete it?

pi-π
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    If $x+y=180^\circ$, what can you say about $\cos(x)$ and $\cos(y)$. More specifically, can you relate $\cos(x)$ and $\cos(180^\circ - x)$ – B. Mehta May 22 '17 at 01:25
  • @labbhattacharjee, How does that question relate to mine? – pi-π May 22 '17 at 01:31
  • Just do it. What is $\cos(180-x)+\cos(x)$? – John Lou May 22 '17 at 01:33
  • @JohnLou, $\cos (180-x)+\cos x=0$. – pi-π May 22 '17 at 01:36
  • Exactly :) How can you write your equation in terms of sets of those? – John Lou May 22 '17 at 01:36
  • @amWhy, isn't the second one with $\sin$ instead of $\cos$? That makes a pretty big difference, I think. – John Lou May 22 '17 at 01:37
  • @JohnLou Yes!! you're right, now removed. Thanks for noticing! – amWhy May 22 '17 at 01:38
  • @JohnLou, No idea. would you please tell? – pi-π May 22 '17 at 01:38
  • @Albert, can't u see the relationship? – lab bhattacharjee May 22 '17 at 01:47
  • @labbhattacharjee, No.. sorry..sir i can't. – pi-π May 22 '17 at 01:48
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    Unfortunately, I can't answer this question anymore, so I'm just gonna write it here because it seems like you need it.

    Your question is equal to $$\sum_{k=1}^{89}(\cos(k)) + \cos(90)+ \sum_{k=91}^{179}(\cos(k)) + \cos(180)$$

    Simplifying down, we get $$\sum_{k=1}^{89}(\cos(k) + \cos(180-k)) + \cos(90) + \cos(180)$$

    Notice how the summation now cancels to zero, leaving us with $$\cos(90) + \cos(180) = 0 + -1 = -1$$

     – John Lou May 22 '17 at 02:12
  • @JohnLou, How did you get the step after " Simplifying down, we get" ? – pi-π May 23 '17 at 01:27
  • Well, look at the two summations. The first term in the first is $1$, the last in the second summation is $179$. $1+179 = 180$. The last term in the first is $89$, the first in the last is $91$. $89+91 = 180$. Therefore, we can see that they are all paired and sum up to 180. – John Lou May 23 '17 at 13:34

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