cos 1^circ + cos 2^circ + cos 3^circ + dots + | vedclass

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Question

(C) The given expression is $S = \cos 1^\circ + \cos 2^\circ + \dots + \cos 179^\circ + \cos 180^\circ$.We know that $\cos(180^\circ - 	heta) = -\cos

Vedclass · Accepted Answer

$-1$

Vedclass · Answer

(C) The given expression is $S = \cos 1^\circ + \cos 2^\circ + \dots + \cos 179^\circ + \cos 180^\circ$.We know that $\cos(180^\circ - 	heta) = -\cos 	heta$.Therefore,$\cos 179^\circ = -\cos 1^\circ$,$\cos 178^\circ = -\cos 2^\circ$,and so on.Pairing the terms from $1^\circ$ to $179^\circ$ as $(\cos 1^\circ + \cos 179^\circ) + (\cos 2^\circ + \cos 178^\circ) + \dots + (\cos 89^\circ + \cos 91^\circ) + \cos 90^\circ$.Since $\cos(180^\circ - 	heta) + \cos 	heta = 0$,each pair sums to $0$.Also,$\cos 90^\circ = 0$.Thus,the sum of the first $179$ terms is $0$.The remaining term is $\cos 180^\circ = -1$.Therefore,the total sum is $0 + (-1) = -1$.

$\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + \dots + \cos 180^\circ = $

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