Question

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

Options

• 0

• 1

• \[\frac{1}{2}\]

   

• \[- \frac{1}{2}\]

   

Answer 1

\[- \frac{1}{2}\]

\[\cos40^\circ + \cos80^\circ + \cos160^\circ + \cos240^\circ\]
\[ = 2\cos\left( \frac{40^\circ + 80^\circ}{2} \right)\cos\left( \frac{40^\circ - 80^\circ}{2} \right) + \cos160^\circ - \cos\left( 180^\circ + 60^\circ \right) \left[ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = 2\cos60^\circ \cos\left( - 20^\circ \right) + \cos160^\circ - \frac{1}{2}\]
\[ = 2 \times \frac{1}{2}\cos20^\circ + \cos160^\circ - \frac{1}{2}\]
\[ = - \cos\left( 180 - 20 \right)^\circ + \cos160^\circ - \frac{1}{2}\]
\[ = - \frac{1}{2}\]