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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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">S</mi><mo form="prefix">sin</mo><mo>*</mo></msubsup></semantics></math></inline-formula>, with the quantity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow><mrow><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mfrac></semantics></math></inline-formula> subordinated to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>+</mo><mo form="prefix">sin</mo><mi>z</mi></mrow></semantics></math></inline-formula>, we obtain an estimate on the initial coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>4</mn></msub></semantics></math></inline-formula> and an upper bound of the third Hankel determinant. For functions in the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">BT</mi><mo form="prefix">sin</mo></msub></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>f</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> lie in an eight-shaped domain in the right-half plane, we prove that its upper bound of third Hankel determinant is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mn>1</mn><mn>16</mn></mfrac></semantics></math></inline-formula>. All the results are proven to be sharp.