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$\cos \theta^\circ + \cos (\theta^\circ+1^\circ) + \cos (\theta^\circ+2^\circ) +\cdots + \cos (\theta^\circ+359^\circ)$

What is the value of this expression when $\theta$ is an integer?

A certain theta value is shown here, but theta can be any integer. Consider how the cosine changes as the angle increases.

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