1. **State the problem:** We have a right triangle CBA with a right angle at B. Side CB is 4.6 units, side CA (the hypotenuse) is 9 units, and we want to find angle $x^\circ$ at vertex C. 2. **Formula used:** In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse: $$\cos(x) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$ 3. **Identify sides:** For angle $x$ at C, the adjacent side is CB = 4.6, and the hypotenuse is CA = 9. 4. **Calculate cosine:** $$\cos(x) = \frac{4.6}{9}$$ 5. **Simplify fraction:** $$\cos(x) = \frac{\cancel{4.6}}{\cancel{9}} = 0.5111...$$ 6. **Find angle $x$ using inverse cosine:** $$x = \cos^{-1}(0.5111)$$ 7. **Calculate and round:** $$x \approx 59.3^\circ$$ **Final answer:** $x \approx 59.3^\circ$