In this paper, the transcendental equation was obtained in the framework of the Euler–Bernoulli beam theory, which allows obtaining the values of resonant frequencies for any tapered nanowire. Calculations of the frequencies of the first few resonances of mechanical oscillations for nanowires with various conicity (the average radius and length of the nanowire remained constant) were performed. It was established that the frequencies of the first three modes increase with an increase in the conicity angle, while the frequency of the fourth mode (n = 4) is nearly constant and independent of the conicity angle. It was also established that the frequencies of the higher order (n > 4) modes decrease with an increase in the conicity angle. The ratios of the resonance values of the first few modes can be used to clarify the conicity value, which is necessary when determining the Young's modulus of tapered nanowires.