1. **Problem Statement:** We have a right triangle PQR with a right angle at P. Side PQ is 33 units, angle Q is 56°, and we need to find side RO (which we interpret as side QR, the hypotenuse) labeled as $x$. 2. **Trig Relationship:** Since angle Q is 56°, side PQ is adjacent to angle Q, and side QR is the hypotenuse. 3. **Formula:** Use the cosine function, which relates adjacent side and hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ Here, $\theta = 56^\circ$, adjacent = 33, hypotenuse = $x$. 4. **Set up the equation:** $$\cos(56^\circ) = \frac{33}{x}$$ 5. **Solve for $x$:** Multiply both sides by $x$ and then divide both sides by $\cos(56^\circ)$: $$x \cdot \cos(56^\circ) = 33$$ $$x = \frac{33}{\cos(56^\circ)}$$ 6. **Intermediate step with cancellation:** $$x = \frac{33}{\cancel{\cos(56^\circ)}} \times \frac{1}{\cancel{\cos(56^\circ)}}$$ (This shows division by $\cos(56^\circ)$ clearly.) 7. **Calculate the value:** $$\cos(56^\circ) \approx 0.5592$$ $$x = \frac{33}{0.5592} \approx 59.0$$ 8. **Final answer:** $$\boxed{x \approx 59.0}$$ This means the hypotenuse side QR is approximately 59.0 units long.