The theorem shown in the image describes the properties of two trigonometric functions:

1. $C(x) = A \cos(\omega x + \phi) + B$
2. $S(x) = A \sin(\omega x + \phi) + B$

These are forms of cosine and sine functions, where:

• $A$ represents the amplitude.
• $\omega$ (omega) represents the angular frequency.
• $\phi$ represents the phase shift.
• $B$ represents the vertical shift.

Why must $\omega > 0$?

1. Angular Frequency and Period: The angular frequency $\omega$ dictates how fast the function oscillates. Specifically, the period of the function, i.e., the time (or interval) it takes for the function to complete one full cycle, is given by: $\text{Period} = \frac{2\pi}{\omega}$ For the period to be a meaningful positive value, $\omega$ must be greater than 0. If $\omega \leq 0$, it would result in an undefined or negative period, which doesn't make sense in the context of oscillatory functions.

2. Interpretation of Negative Frequency: If $\omega$ were negative, it would imply the function is oscillating in the opposite direction. In practice, we use positive values of $\omega$ to represent forward oscillations and simply adjust the phase shift $\phi$ to account for direction or starting position.

3. Behavior of Zero Frequency: If $\omega = 0$, the cosine or sine function becomes constant because: $\cos(0 \cdot x + \phi) = \cos(\phi), \quad \sin(0 \cdot x + \phi) = \sin(\phi)$ In this case, the function no longer oscillates, and the system would only have a constant value depending on $\phi$, losing the periodic behavior entirely.

In summary, $\omega > 0$ ensures that the trigonometric functions exhibit regular oscillatory behavior with a well-defined period. If $\omega$ were zero or negative, the behavior would either become constant or reverse, which isn't typical in standard oscillation theory.

Would you like further explanation on any of these points?

Here are some follow-up questions to deepen your understanding:

1. How does the value of $\omega$ affect the speed of oscillations?
2. What happens to the function when the phase shift $\phi$ changes?
3. Can you describe the significance of amplitude $|A|$ in these functions?
4. How does the vertical shift $B$ alter the function’s graph?
5. What real-world phenomena can be modeled by these types of sine and cosine functions?

Tip: If you encounter oscillatory systems in physics, like waves or alternating currents, the concept of angular frequency $\omega$ and its relationship to the period is essential for understanding periodic motion.