1. **State the problem:** We have a right triangle with an angle of $49.2^\circ$ at vertex A, the side opposite this angle is 18 units, and we want to find the lengths of the adjacent side and the hypotenuse. 2. **Relevant formulas:** In a right triangle, the sine, cosine, and tangent functions relate the angles to the sides: - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Find the hypotenuse:** Using sine: $$\sin(49.2^\circ) = \frac{18}{\text{hypotenuse}}$$ Rearranged: $$\text{hypotenuse} = \frac{18}{\sin(49.2^\circ)}$$ Calculate $\sin(49.2^\circ) \approx 0.7557$: $$\text{hypotenuse} = \frac{18}{0.7557} \approx 23.81$$ 4. **Find the adjacent side:** Using tangent: $$\tan(49.2^\circ) = \frac{18}{\text{adjacent}}$$ Rearranged: $$\text{adjacent} = \frac{18}{\tan(49.2^\circ)}$$ Calculate $\tan(49.2^\circ) \approx 1.1504$: $$\text{adjacent} = \frac{18}{1.1504} \approx 15.64$$ 5. **Summary:** - Hypotenuse $\approx 23.81$ - Adjacent side $\approx 15.64$ These values satisfy the Pythagorean theorem and the trigonometric ratios for the given angle.