Mathematico | Mathematico

Find the value of

\sin (1) \cos (359) + \sin (2) \cos (358) + \dots + \sin (359) \cos (1)

One way to do this is to arrange the terms so that the first and last terms are paired together:

\begin{aligned} \sin (1) \cos (359) + \sin (359) \cos (1) \\ + & \sin (2) \cos (358) + \sin (358) \cos (2) \\ ⋮ \\ + & \sin (179) \cos (181) + \sin (181) \cos (179) \\ + & \sin (180) \cos (180) \end{aligned}

Compare $\sin (1) \cos (359) + \sin (359) \cos (1)$ with

\sin (A + B) = \sin A \cos B + \cos A \sin B