Solution: The hypotenuse is $\sqrt{9^2 + 12^2} = 15$ cm. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15} = \frac{3}{5}$.

["Understanding Right Triangles: How to Find $\sin \ heta$ with the Hypotenuse Calculated", "When solving problems in trigonometry involving right triangles, one of the most essential concepts is finding the sine of an angle using the ratio of the opposite side to the hypotenuse. In many geometry and physics applications, calculating the hypotenuse correctly is the first step to uncovering key trigonometric values.", "Consider a right triangle where one leg measures 9 cm and the other leg measures 12 cm. The hypotenuse, found using the Pythagorean theorem, becomes:", "[\n\ ext{Hypotenuse} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \ ext{ cm}\n]", "This precise calculation confirms the triangle’s geometry and sets the foundation for angular ratios. The sine of angle θ—defined as the ratio of the opposite side to the hypotenuse—follows immediately:", "[\n\sin \ heta = \frac{\ ext{opposite}}{\ ext{hypotenuse}} = \frac{9}{15} = \frac{3}{5}\n]", "This value, $\frac{3}{5}$, not only simplifies cleanly but also represents a fundamental trigonometric truth within a right triangle with integer side lengths—often called a 3-4-5 triangle scaled by a factor of 3.", "Understanding how to derive the hypotenuse first ensures accuracy in further trigonometric computations. Whether used in physics problems involving forces, engineering designs, or classroom geometry exercises, knowing $\sin \ heta = \frac{3}{5}$ opens the door to deeper insight into right triangle relationships.", "Key takeaway: Double-checking your hypotenuse using $c = \sqrt{a^2 + b^2}$ establishes a solid base for computing all trigonometric ratios. In a triangle with legs 9 cm and 12 cm, this method confirms the hypotenuse as 15 cm and $\sin \ heta = \frac{3}{5}$—a core skill every learner and practitioner should master.", "---", "Keywords: hypotenuse calculation, $\sin \ heta$, right triangle trigonometry, Pythagorean theorem, 3-4-5 triangle, trigonometric ratios, geometry problems, sine function, $\frac{3}{5}$, $\sin \ heta = \frac{opposite}{hypotenuse}$, right triangle sine."]