Wavelet is a small wave, what is problem with using sin(x) of one cycle?
A 1-cycle sinewave wavelet corresponds to imaginary component basis vector for bin 1 of a rectangularly windowed DFT result, and thus has the same problems of the lowest bins near DC of a rectangular windowed FFT.  

The frequency response of a rectangular windowed FFT is a Sinc function with an expanse of large ripples in the response. The main lobe of the frequency response would be very wide relative to the frequency of that wavelet's basis. And the complex conjugate image, or negative frequency response, is nearby to the lowest bin(s); and thus their side lobes can potentially cause interference with the positive frequency response, depending on phase.

e.g. the wavelet would be highly subject to interference  from noise and any other spectral content that wasn’t purely sinusoidal and exactly integer periodic relative to the wavelets length.