Question

$\cos {1^ \circ } + \cos {2^ \circ } + \cos {3^ \circ } + ............ + \cos {180^ \circ }$ is equal to
1. $1$
2. $0$
3. $2$
4. $ - 1$

Answer 1

Hint: Add at least four values in the last of the given sum. Apply trigonometric sign convention formula basically to make angles negative and eradicate some of them to minimize the values and easy to find the sum.

Complete step by step solution:  Given that,
$ = \cos {1^ \circ } + \cos {2^ \circ } + \cos {3^ \circ } + ............ + \cos {180^ \circ }$
$ = \cos {1^ \circ } + \cos {2^ \circ } + \cos {3^ \circ } + ............ + \cos {177^ \circ } + \cos {178^ \circ } + \cos {179^ \circ } + \cos {180^ \circ }$
$ = \cos {1^ \circ } + \cos {2^ \circ } + \cos {3^ \circ } + .... + \cos {89^ \circ } + \cos {90^ \circ } + \cos \left( {{{180}^ \circ } - {{91}^ \circ }} \right) + \cos \left( {{{180}^ \circ } - {{92}^ \circ }} \right) + ........ + \cos \left( {{{180}^ \circ } - {3^ \circ }} \right) + \cos \left( {{{180}^ \circ } - {2^ \circ }} \right) + \cos ({180^ \circ } - {1^ \circ }) + \cos {180^ \circ }$
$ = \cos {1^ \circ } + \cos {2^ \circ } + \cos {3^ \circ } + .... + \cos {89^ \circ } + \cos {90^ \circ } + \left( { - \cos {{89}^ \circ }} \right) + \left( { - \cos {{88}^ \circ }} \right) + ........ + \left( { - \cos {3^ \circ }} \right) + \left( { - \cos {2^ \circ }} \right) + \left( { - \cos {1^ \circ }} \right) + \cos {180^ \circ }$
Cancel all the like terms in the above equation
$ = \cos {90^ \circ } + \cos {180^ \circ }$
$ = 0 + ( - 1)$
$ = - 1$
Hence, the sum of Given terms $\cos {1^ \circ },\cos {2^ \circ },\cos {3^ \circ },............,\cos {180^ \circ }$ is equals to $ - 1$
$ \therefore$ Option (4) is the correct answer i.e., $ - 1$.

Note: While using the sign convention trigonometric formulas don’t get confused with the values make the sign convention graph then apply the formula and use correct signs carefully or keep the table of all the formulas with you and try to learn them.