1. **State the problem:** We are given a right triangle with angle $\theta$, the side opposite $\theta$ is 6, and the hypotenuse is 10. We need to find $\sin \theta$. 2. **Recall the definition of sine:** $$\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$$ 3. **Apply the formula:** $$\sin \theta = \frac{6}{10}$$ 4. **Simplify the fraction:** $$\sin \theta = \frac{3}{5}$$ 5. **Interpretation:** The sine of angle $\theta$ is $\frac{3}{5}$, which means the ratio of the length of the side opposite $\theta$ to the hypotenuse is $\frac{3}{5}$. **Final answer:** $$\sin \theta = \frac{3}{5}$$