THREE-PHASE CURRENT Show that, for any time \(t\). \(\cos \omega t+\cos (\omega t+2 \pi / 3)+\cos (\omega t+4 \pi / 3)=0\).

The cosine addition formula is a fundamental tool in trigonometry that relates the cosine of a sum of two angles to the cosines and sines of the individual angles. The formula is stated as \( \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta \). This formula is essential for solving the exercise at hand, which involves showing that the sum of three cosine functions, each offset by a phase shift, equals zero.

Particularly in the context of three-phase currents, we leverage the cosine addition formula to handle the angular shifts typical of three-phase systems. By applying this formula, and substituting the values of \( \alpha = \omega t \) and \( \beta \) as either \( \frac{2\pi}{3} \) or \( \frac{4\pi}{3} \) we can simplify complex expressions down to more manageable terms that aid in demonstrating the balance of three-phase systems.

Trigonometric identities are equations involving trigonometric functions that hold true for all allowed values of the occurring variables. Commonly used trigonometric identities include those for the sine and cosine of particular angles, which often lead to significant simplifications in various mathematical problems, especially those concerning periodic functions, like those found in the analysis of alternating currents.

In our three-phase current problem, after applying the cosine addition formula, we use identities such as \( \cos(\frac{2\pi}{3}) = -\frac{1}{2} \) and \( \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} \) to find the values required for the summation step. These specific values arise due to the symmetrical nature of the trigonometric functions on the unit circle. Recognizing and using these identities correctly allows us to transform the original complex expression into a simple equation that can be easily proven to be true.

Electromagnetic theory encompasses the laws of electricity and magnetism and how they unify to form electromagnetic fields, which are fundamental to understanding not just three-phase current, but also a wide range of phenomena in physics and engineering disciplines. Electromagnetic fields are pivotal in power generation, transmission, and distribution systems.

In the context of three-phase electric power systems, the electromagnetic theory comes into play in the generation and functioning of alternating currents that are out of phase with one another, typically by 120 degrees, yielding a \( \frac{2\pi}{3} \) phase shift. This is the root of the exercise's phase offsets. Understanding the principles of electromagnetic theory helps explain not just why these systems are efficient in power delivery and use, but it also underlies the mathematical derivations that describe system behavior, such as equations involving three-phase current. Demonstrating the zero-sum relationship among the three phases highlights the balance and symmetry inherent in such systems, which is a direct consequence of electromagnetic principles.