We study the problem of constructing designs that are optimal for estimating individual coefficients in a trigonometric regression model without an intercept on [-a, a]. We provide a complete solution of the e(k)-optimality problem. On the full circle [-pi, pi], optimal designs coincide with their counterparts for the model with an intercept. For partial circles (a < r), sine-coefficient designs are shown to be inherited from the intercept model. For cosine coefficients, we study the problem of constructing e(k)-optimal designs in a trigonometric regression model without an intercept on [-a, a]. For cosine coefficients, we derive an explicit class of designs that provide highly efficient approximations to the optimal ones for a up to a threshold a(max). A key finding is that the designs are strictly optimal for m = 2, while for m > 2 they remain nearly 100% efficient, with the range of practical validity expanding as the model order m increases.