AC Circuit Help

[Loki]

Can anyone help me with this homework problem? I worked it through, but the answer i get doesn't agree with the answer the book gives. I'll include my work, and if anyone can help, I'll be eternally grateful, at least for a while.

The Problem:

In a series RCL circuit the dissapated power drops by a factor of two when the frequency of the generator is changed from the resonant frequency to a nonresonant frequency. The peak voltage is held constant while this change is made. Determine the power factor of the curcuit at nonresonant frequency.

My Work:

P_r = 2 P_{nr}
P = I * V * \cos \phi
\phi = \arctan \frac{X_L - X_C}{R}
At resonant frequency, X_L - X_C = 0
I * V * \cos ( \arctan \frac{X_L-X_C}{R} ) = 2 * I * V \cos (\arctan\frac{X_L-X_C}{R})
I * V * \cos ( \arctan (0)) = 2 * I * V \cos (\arctan\frac{X_L-X_C}{R})
I * V * \cos (0) = 2 * I * V \cos (\phi)
1 = 2 \cos (\phi)
\cos \phi = \frac{1}{2}

This seems correct to me, but the book says the answer is \frac{\sqrt {2}}{2}

Am I mistaking the "Factor of two" part of the problem? Does this really mean that the power is squared?


If anybody can help, thanks. On a sidenote, LaTeX is very nice, if a bit tedious to write in.