1. **State the problem:** We have a right triangle with angle $U = 66^\circ$, side $ST = 7.5$, and side $UT = x$. Angle $T$ is the right angle. We need to solve for $x$. 2. **Identify the sides relative to angle $U$:** - Side $ST = 7.5$ is the hypotenuse (opposite the right angle). - Side $UT = x$ is the side adjacent to angle $U$. 3. **Choose the trigonometric function:** Since we know the hypotenuse and want the adjacent side, we use cosine: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 4. **Write the equation:** $$\cos(66^\circ) = \frac{x}{7.5}$$ 5. **Solve for $x$:** $$x = 7.5 \times \cos(66^\circ)$$ 6. **Calculate the cosine:** $$\cos(66^\circ) \approx 0.4067$$ 7. **Find $x$:** $$x = 7.5 \times 0.4067 = 3.05025$$ 8. **Round to the nearest tenth:** $$x \approx 3.1$$ **Final answer:** $x \approx 3.1$