Question

Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.

Answer 1

$$\begin{gathered} \cos 1°+\cos 2°+\cos 3°+...+\cos 180° \\ =\cos 1°+\cos 2°+\cos 3°+...+\cos 88°+\cos 89°+\cos 90°+\cos \left(180-89\right)°+\cos \left(180-88\right)°+...+\cos \left(180-1\right)°+\cos 180°\left[\cos \left(180°-\theta \right)=-\cos \theta \right] \\ =\cos 1°+\cos 2°+\cos 3°+...+\cos 88°+\cos 89°+\cos 90°-\cos 89°-\cos 88°-...-\cos 1°+\cos 180° \\ =\cos 90°+\cos 180° \\ =0-1 \\ =-1 \end{gathered}$$