Write the value of 1^ + 2^ + 3^ + ... + 180^ . - Vidyadip

Question

Write the value of $\cos1^\circ+\cos2^\circ+\cos3^\circ+\ ...\ +\cos180^\circ.$

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Answer

$\cos(180^\circ-\theta)=-\cos\theta$ $\cos1^\circ+\cos2^\circ+\ ...\ +\cos180^\circ$ $=\cos(180^\circ-179^\circ)+\cos(180^\circ-178^\circ)+\ ...\ +\cos90^\circ+\cos91^\circ+\ ...\ +\cos180^\circ$ $=-\cos179^\circ-\cos178^\circ-\ ...\ -\cos91^\circ+\cos90^\circ+\cos91^\circ+\ ...\ +\cos180^\circ$ $=\cos90^\circ+\cos180^\circ$ $=0-1$ $=-1$

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