z-transform of sin kθ and aᵏ sin kθ

Z-transform of sin kθ

sin kθ = (e^{jkθ} − e^{−jkθ})/(2j), and Z[bᵏ] = z/(z − b).

Z[sin kθ] = (1/2j) [ z/(z − e^{jθ}) − z/(z − e^{−jθ}) ]
          = (1/2j) · z (e^{jθ} − e^{−jθ}) / (z² − 2z cos θ + 1)
          = (1/2j) · z (2j sin θ) / (z² − 2z cos θ + 1)

Z[sin kθ] = z sin θ / (z² − 2z cos θ + 1), |z| > 1.

Z-transform of aᵏ sin kθ

By the scaling property, Z[aᵏ x(k)] = X(z/a). Replace z by z/a:

Z[aᵏ sin kθ] = (z/a) sin θ / ( z²/a² − 2(z/a) cos θ + 1 )
             = a z sin θ / ( z² − 2a z cos θ + a² )

Answer: Z[aᵏ sin kθ] = a z sin θ / (z² − 2a z cos θ + a²), |z| > |a|.

(Numerical check, a = 0.5, θ = 0.7, z = 3: sum of series = 0.138930, formula = 0.138930.)