\( S = \cos ( 2 t ) \sin ( 4 t ) \)

Did you know that the function \( S = \cos(2t) \sin(4t) \) can be simplified using a neat trigonometric identity? Yep! You can use the product-to-sum identities, which state that \( \cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)] \). For our function, that means \( S = \frac{1}{2} [\sin(6t) - \sin(2t)] \). Who knew trigonometry could be so handy? Now, when you’re working with trigonometric functions like \( S \), keep an eye out for common mistakes, like forgetting to respect the periodic nature of sine and cosine. Those sneaky factors can lead you to miscalculate the amplitude or create confusion in determining the wave’s shift. Don’t fret though! Just remember to sketch the graph or check the phase relationships!