Trigonometry - Trigonometric Graphs and Parameters

A radio device transmits signals by electromagnetic waves, where the electric field of the waves can be expressed as \(\text{E}\hspace{0.1em}(t) = E_0 \cos(2\pi f t)\) (in volt, \(\text{V}\)), with respect to time \(t\) (in seconds). \(f\) is called the frequency of the wave in hertz (\(\text{Hz}\)). Assume that the amplitude of the electric field is \(5\) volt, the frequency of the electric field is \(100\) MHz (\(1\text{ MHz} = 10^6\text{ Hz}\)).

1. Find the specific expression of the electric field \(\text{E}\hspace{0.1em}(t)\) of this electromagnetic wave.
2. Find the period of \(\text{E}\hspace{0.1em}(t)\). Hence derive the relationship between frequency and period.
3. Sketch the graph of \(\text{E}\hspace{0.1em}(t)\), for \(0 \le t \le 4 \times 10^{-8}\).
4. In different circumstances, the graphs of \(\text{E}\hspace{0.1em}(t)\) are shown in the figures below, respectively. Briefly describe the possible operations on the radio device in terms of power and/or frequency tuning in each of the cases.