1. **State the problem:** Given the opposite side length $20$ and an angle of $59^\circ$, find the length of the adjacent side $q$ in a right triangle. 2. **Formula used:** In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Apply the formula:** Substitute $\theta = 59^\circ$ and opposite $= 20$: $$\tan(59^\circ) = \frac{20}{q}$$ 4. **Solve for $q$:** Multiply both sides by $q$ and then divide both sides by $\tan(59^\circ)$: $$q \times \tan(59^\circ) = 20$$ $$q = \frac{20}{\tan(59^\circ)}$$ 5. **Calculate the value:** Using a calculator, $$\tan(59^\circ) \approx 1.6643$$ So, $$q = \frac{20}{1.6643}$$ 6. **Simplify:** $$q \approx 12.02$$ **Final answer:** The length of the adjacent side $q$ is approximately $12.02$ units.