Question

Sa se calculeze cos0+cos1+cos2+...+cos180 grade

Answer 1

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Answer 2

$\cos0^{\circ}+\cos1^{\circ}+\cos2^{\circ}+...+\cos180^{\circ} = \\ \\ = \cos180^{\circ}+\cos179^{\circ}+...+\cos90^{\circ} +...+\cos2^{\circ}+\cos1^{\circ}+\cos0^{\circ} = \\ \\ = \cos180^{\circ}+\cos(180^{\circ}-1^{\circ})+\cos (180^{\circ}-2^{\circ})+...+\cos(180-89)+ \\ +\cos90^{\circ}+\cos89^{\circ}+\cos88^{\circ}+...+\cos2^{\circ}+\cos1^{\circ}+\cos0^{\circ} = \\ \\ = -1-\cos1^{\circ}-\cos2^{\circ}-...-\cos89^{\circ}+0+\cos89^{\circ}+\cos88^{\circ}+...+ \\ +\cos2^{\circ}+\cos1^{\circ}+1 =$

$=-1 + (\cos1^{\circ}+\cos2^{\circ}+...+\cos89^{\circ})-(\cos1^{\circ}+\cos2^{\circ}+...+\cos89^{\circ})+1 = \\ \\ = -1+1 \\ \\ = 0$

$\\\boxed{ M-am folosit de identitatea: \quad \cos(180^\circ-x) = -\cos x}$