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Question
Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.
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Solution
\[\cos1^\circ + \cos2^\circ + \cos3^\circ + . . . + \cos180^\circ\]
\[ = \cos1^\circ + \cos2^\circ + \cos3^\circ + . . . + \cos88^\circ + \cos89^\circ + \cos90^\circ + \cos\left( 180 - 89 \right)^\circ + \cos\left( 180 - 88 \right)^\circ + . . . + \cos\left( 180 - 1 \right)^\circ + \cos180^\circ \left[ \cos\left( 180^\circ - \theta \right) = - \cos \theta \right]\]
\[ = \cos1^\circ + \cos2^\circ + \cos3^\circ + . . . + \cos88^\circ + \cos89^\circ + \cos90^\circ - \cos89^\circ - \cos88^\circ - . . . - \cos1^\circ + \cos180^\circ\]
\[ = \cos90^\circ + \cos180^\circ\]
\[ = 0 - 1\]
\[ = - 1\]
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