Solveeit Logo
Login

Question

Question: 1 cos 40° + cos 80° + cos 160° + cos 240° =...

1 cos 40° + cos 80° + cos 160° + cos 240° =

A

1

B
  • 2
C
  • 1/2
D

0

Answer

-1/2

Explanation

Solution

To evaluate cos⁡40∘+cos⁡80∘+cos⁡160∘+cos⁡240∘\cos40^\circ + \cos80^\circ + \cos160^\circ + \cos240^\circ:

  • Use the identity: cos⁡40∘+cos⁡80∘=2cos⁡60∘cos⁡20∘=2⋅12cos⁡20∘=cos⁡20∘\cos40^\circ + \cos80^\circ = 2\cos60^\circ\cos20^\circ = 2\cdot\frac{1}{2}\cos20^\circ = \cos20^\circ.

  • Note that: cos⁡160∘=cos⁡(180∘−20∘)=−cos⁡20∘\cos160^\circ = \cos(180^\circ-20^\circ) = -\cos20^\circ cos⁡240∘=cos⁡(180∘+60∘)=−cos⁡60∘=−12\cos240^\circ = \cos(180^\circ+60^\circ) = -\cos60^\circ = -\frac{1}{2}.

Therefore,

cos⁡40∘+cos⁡80∘+cos⁡160∘+cos⁡240∘=cos⁡20∘−cos⁡20∘−12=−12\cos40^\circ + \cos80^\circ + \cos160^\circ + \cos240^\circ = \cos20^\circ - \cos20^\circ -\frac{1}{2} = -\frac{1}{2}.