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

Question:

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

(a) 0

(b) 1

(c) $\frac{1}{2}$

(d) $-\frac{1}{2}$

Solution:

(d) $-\frac{1}{2}$

$\cos 40^{\circ}+\cos 80^{\circ}+\cos 160^{\circ}+\cos 240^{\circ}$

$=2 \cos \left(\frac{40^{\circ}+80^{\circ}}{2}\right) \cos \left(\frac{40^{\circ}-80^{\circ}}{2}\right)+\cos 160^{\circ}-\cos \left(180^{\circ}+60^{\circ}\right)$

$\left[\because \cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$

$=2 \cos 60^{\circ} \cos \left(-20^{\circ}\right)+\cos 160^{\circ}-\frac{1}{2}$

$=2 \times \frac{1}{2} \cos 20^{\circ}+\cos 160^{\circ}-\frac{1}{2}$

$=-\cos (180-20)^{\circ}+\cos 160^{\circ}-\frac{1}{2}$

$=-\frac{1}{2}$

Leave a comment

Close

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now