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A new algorithm is given for computing trigonometric functions using the logarithmic number system (LNS). Based on the laws of sines and cosines, the algorithm uses novel addressing of ROMs with the middle-order bits of the LNS representation. Error analysis and simulation show the algorithm is accurate to 22 bits when intermediate steps are performed with 23-bit precision LNS. A VLIW software implementation having throughput of one trigonometric result every 17 cycles is suggested that uses special instructions to access small ROMs containing logarithmic sines and cosines. Also, the proposed algorithm can be implemented fully in hardware having throughput of one trigonometric result every one or two cycles using minor low-cost modifications to an existing LNS ALU design.