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The main focus of this research is to solve certain coefficient-related problems for analytic functions that are subordinated to a unique trigonometric function. For the class Ssin*, with the quantity zf′(z)f(z) subordinated to 1+sinz, we obtain an estimate on the initial coefficient a4 and an upper bound of the third Hankel determinant. For functions in the class BTsin, with f′(z) lie in an eight-shaped domain in the right-half plane, we prove that its upper bound of third Hankel determinant is 116. All the results are proven to be sharp.

The Sharp Bounds of the Third-Order Hankel Determinant for Certain Analytic Functions Associated with an Eight-Shaped Domain