Question

Work the correct alternative in each of the following
1. cos 40° + cos 80° + cos 160° + cos 240° =​

Answer 1

Question :

find the value of

cos40° + cos 80° + cos 160° + cos 240°

Formulas used :

1. cos (180-x ) = -cosx
2. cos (180+ x) = -cosx
3. $cosx + cosy \: = 2 \cos( \frac{x + y}{2} ) \cos( \frac{x - y}{2} )$
4. cos(-x) = cosx

Solution :

cos40° + cos 80° + cos 160° + cos 240°

⇒cos40° + cos 80° + cos (180 °-20°)+ cos (180°+60°)

[ use formula : cos (180- x) = -cosx

and cos(180+ x) = -cosx ]

⇒cos 40° + cos 80° - cos 20° - cos 60°

apply formula cos x+ cos y

________________________

$\implies \: 2 \cos( \frac{40 + 80}{2} ) \cos( \frac{40 - 80}{2} ) - \cos(20) \degree - \cos(60) \degree$

⇒2 cos 60° cos 20° - cos 20° - cos 60°

put the value of cos60°

_______________________

$\implies \: 2 \times \frac{1}{2} \cos(20) \degree \: - \cos(20) \degree \: - \cos(60) \degree$

⇒cos20° - cos 20° - cos 60°

⇒ - cos 60°

$\implies \: - \cos(60) \degree = - \frac{1}{2}$

Required Answer :$\frac{-1}{2}$