Possible excitation of thermonuclear cone instabilities caused by anisotropy in alpha-particle velocity distribution is considered in tokamak reactor plasmas for frequencies much lower than electron cyclotron frequency. In the ignited tokamak reactor experiment, for example the International Thermonuclear Experimental Reactor (ITER), Alfvén waves are excited at frequencies comparable to alpha-particle cyclotron frequency. The growth rate, the region of instability localization in the plasma cross section, and the marginal stability boundaries in the n-T plane are determined for cone Alfvén wave instability.