Question

Find the value of given trigonometric expression
Cos40 + cos80 + cos160 + cos240
  (A) 0
  (B) 1
  (C) $\frac{1}{2}$
  (D) $ - \frac{1}{2}$

Answer 1

Hint- In this type of questions , convert the cosines by checking the sign of them by using the quadrant in which they lie and use the formula for cosC + cosD to get to the desired answer . By using the supplementary angles of cosine to find the value of cos(160) and cos(240).
Complete step-by-step solution -
$\cos 40 + \cos 80 + \cos (180 - 20) + \cos (180 + 60)$
$ \Rightarrow 2\cos \left( {\dfrac{{80 + 40}}{2}} \right)\left( {\dfrac{{80 - 40}}{2}} \right) - \cos 20 - \cos 60$ ( Since , \[\operatorname{cosC} + cosD = 2\cos \dfrac{{C + D}}{2} \times \cos \dfrac{{C - D}}{2}\] )

 (Also , $\cos \left( {180 - \theta } \right) = - \cos \theta ,\cos \left( {180 + \theta } \right) = - \cos \theta $ , cos$\theta $ becomes negative on second and third quadrant)

$ \Rightarrow 2 \times \cos 60 \times \cos 20 - \cos 20 - \cos 60$

$ \Rightarrow 2 \times \dfrac{1}{2} \times \cos 20 - \cos 20 - \cos 60 = - \cos 60 = - \dfrac{1}{2}$

Note- Remember the basics about the signs of trigonometric functions in different quadrants to solve such kinds of questions. Use the correct trigonometric formulas to make the solution simple and accurate.