\( \sin (3x)+\sin (x) \)

Did you know that the expression \( \sin(3x) \) can be transformed using the angle addition formulas? By applying the triple angle formula, you can rewrite it as \( 3\sin(x) - 4\sin^3(x) \). So the overall expression becomes \( 3\sin(x) - 4\sin^3(x) + \sin(x) = 4\sin(x) - 4\sin^3(x) \). Now that’s what I call a transformation party! In real-world applications, sine functions often pop up in fields like engineering, physics, and music! For instance, when analyzing waves, \( \sin(x) \) can help model sound waves or light waves, showcasing how these functions are foundational in understanding various oscillatory phenomena. So whether you’re designing a roller coaster or tuning a piano, sine waves are everywhere—rocking your world!