1. **State the problem:** We have a right triangle with a hypotenuse of length 19, an angle of 74°, and we need to find the length of the side opposite the 74° angle, labeled $x$. 2. **Formula used:** In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ 3. **Apply the formula:** Here, $\theta = 74^\circ$, opposite side = $x$, hypotenuse = 19. $$\sin(74^\circ) = \frac{x}{19}$$ 4. **Solve for $x$:** Multiply both sides by 19: $$x = 19 \times \sin(74^\circ)$$ 5. **Calculate the sine value:** Using a calculator, $$\sin(74^\circ) \approx 0.9613$$ 6. **Find $x$:** $$x = 19 \times 0.9613 = 18.265$$ 7. **Final answer:** The length of the side opposite the 74° angle is approximately $$\boxed{18.27}$$