\( \sin (a+b-c)+\sin (2a)+\sin (a-b+c) \)

Did you know that the sine function, which you’re working with here, has roots in ancient civilizations? The Babylonians had their own versions of trigonometric tables as early as 300 BC, and Indian mathematicians like Aryabhata made significant contributions to trigonometry by defining the sine function long before it became widely popular in the West! Now, let’s talk about this expression you’re dealing with. To simplify or solve it, a common mistake is to overlook the addition formulas for sine. When combining angles, don’t forget to apply those wonderfully intricate sine addition identities properly; they can help streamline your calculations and lead to a clearer, more manageable expression. Enjoy working through the fun of those identities!