1. **State the problem:** We have a right triangle RQP with a right angle at Q, angle R measuring 19°, side RQ = 7.6, and we need to find side RP = x. 2. **Identify the sides relative to angle R:** - Side RQ (7.6) is adjacent to angle R. - Side RP (x) is the hypotenuse. 3. **Use the cosine function:** The cosine of an angle in a right triangle is the ratio of the adjacent side over the hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 4. **Set up the equation:** $$\cos(19^\circ) = \frac{7.6}{x}$$ 5. **Solve for x:** Multiply both sides by $x$: $$x \cos(19^\circ) = 7.6$$ Divide both sides by $\cos(19^\circ)$: $$x = \frac{7.6}{\cos(19^\circ)}$$ Intermediate step showing cancellation: $$x = \frac{7.6}{\cancel{\cos(19^\circ)}} \times \frac{1}{\cancel{\cos(19^\circ)}}$$ 6. **Calculate the value:** $$\cos(19^\circ) \approx 0.9455$$ $$x \approx \frac{7.6}{0.9455} \approx 8.0$$ 7. **Final answer:** $$\boxed{8.0}$$ The length of side RP is approximately 8.0 units.