Chapter 6.4, Problem 89AYU

Alternating Current (ac) Generators The voltage $\text{V}$ produced by an ac generator is sinusoidal. As a function of time, the voltage $\text{V}$ is

$V\left(t\right){\text{ = V}}_{\text{0}}\sin \left(2\pi ft\right)$

where $\text{f}$ is the frequency, the number of complete oscillations (cycles) per second. [In the United States and Canada, $\text{f}$ is 60 hertz (Hz).] The power $P$ delivered to a resistance $R$ at any time $t$ is defined as

$P\left(t\right)\text{ = }\frac{{\left[V\left(t\right)\right]}^2}{R}$

(a) Show that $P\left(t\right)\text{ = }\frac{V_0^2}{R}{\sin }^2\left(2\pi ft\right)$.

(b) The graph of $P$ is shown in the figure. Express Pas a sinusoidal function.

(c) Deduce that

${\sin }^2\left(2\pi ft\right)\text{ = }\frac{1}{2}\left[1-\text{ cos}\left(4\pi ft\right)\right]$