Consider the signal x(t) = 4 cos
 2π
3 t + π
3

.
(a) What is the period of this signal?
(b) What is the value of this signal at t = 0, i.e., what is x(0)?
(c) Plot the signal

Here is the plot of the signal \( x(t) = 4 \cos \left( 2\pi \frac{1}{3} t + \frac{\pi}{3} \right) \), showing its oscillation over time. You can observe the periodic nature of the signal, with a period of 3 seconds, as calculated earlier.

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions that expand on this concept:
1. How would the period of the signal change if the frequency were doubled?
2. What would the plot look like if the phase shift were \( \frac{\pi}{2} \) instead of \( \frac{\pi}{3} \)?
3. How do you determine the amplitude of a cosine signal from its equation?
4. How does adding a sine component to the signal change its behavior?
5. What is the relationship between frequency and angular velocity in a cosine signal?

**Tip:** In periodic signals, the phase shift \( \phi \) affects the horizontal translation but doesn't change the signal's frequency or amplitude.

Consider the signal x(t) = 4 cos(2π/3 * t + π/3).
(a) What is the period of this signal?
(b) What is the value of this signal at t = 0, i.e., what is x(0)?
(c) Plot the signal.

Learn how to determine the period and initial value of the cosine signal x(t) = 4 cos(2π/3 * t + π/3). This problem covers key trigonometric concepts, such as period calculation and phase shifts, and includes a detailed plot of the signal over time. Suitable for students in Grades 10-12.

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