Question

The value of $cos20{}^{\circ }+cos100{}^{\circ }+cos140{}^{\circ }$ is equal to

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\sqrt{3}$
• D. $0$
• E. $1$

Answer 1

Answer: (D)

$cos20{}^{\circ }+cos100{}^{\circ }+cos140{}^{\circ }$
$\Rightarrow$ $cos20{}^{\circ }+cos(90{}^{\circ }+10{}^{\circ })+cos(90{}^{\circ }+50{}^{\circ })$
$=cos20{}^{\circ }-sin10{}^{\circ }-sin50{}^{\circ }$
$=cos20{}^{\circ }-(sin10{}^{\circ }+sin50{}^{\circ })$
$=cos20{}^{\circ }-2sin30{}^{\circ }cos20{}^{\circ }$ $(\because \sin C+\sin D=2\sin \frac{(C+D)}{2}.\cos \frac{(C-D)}{2})$
$=\cos 20{}^{\circ }-2.\frac{1}{2}\cos 20{}^{\circ }$
$=\cos 20{}^{\circ }-\cos 20{}^{\circ }=0$