The relativistic Doppler-shifted cyclotron resonance condition for electromagnetic wave interactions with a plasma is studied. The results indicate that the resonance condition defines an ellipse in velocity space when the product of the index of refraction and cosine of the wave normal angle is less than or equal to 1 and that it defines a partial ellipse when the product is larger than 1. It is also shown that waves with frequencies greater than the gyrofrequency (or its harmonic for higher harmonic resonances) can only resonate with particles moving in the same direction along the magnetic field, while waves with frequencies less than the gyrofrequency (or its harmonic) resonator with particles moving in both direction along the magnetic field. Growth rates for the terrestrial auroral kilometric and Z mode radiation have been calculated by using the relativistic resonance condition and the electron pitch angle distribution measured by the S5-3 satellite in the auroral zone region. In the case of auroral kilometric radiation, it is shown that both the upgoing and the downgoing electrons are unstable and can give rise to the growth of auroral kilometric radiation, although growth due to the downgoing electrons is limited to a much narrower frequency band. The magnitudes of the growth rates for both the ungoing and the downgoing auroral kilometric radiation are comparable and they indicate that the path lengths required to account for the observed intensities of auroral kilometric radiation are of the order of a few hundred kilometers, which are probably too large. Therefore in order for cyclotron maser instability to remain a viable mechanism for the generation of auroral kilometric radiation, it is essential to either have steeper slopes (∂F/∂v⊥) in the loss cone or reflections to increase the path length. The growth rate calculations for the Z mode radiation show that for wave frequencies just below the gyrofrequency and wave normal angles at or near 90¿, the electron distribution is unstable and the growth rates are large enough to account for the observed intensities. The fact that Z mode waves with frequencies smaller than the gyrofrequency are amplified can explain the DE 1 observations of Z mode radiation which indicate no radiation exists above the local electron gyrofrequency.