1. **State the problem:** Given a triangle with sides $a=20$, $b=18$, and angle $C=75^\circ$, find side $c$ using the Law of Cosines. 2. **Formula:** The Law of Cosines states: $$c^2 = a^2 + b^2 - 2ab \cos C$$ This formula relates the lengths of the sides of a triangle to the cosine of one of its angles. 3. **Substitute known values:** $$c^2 = 20^2 + 18^2 - 2 \times 20 \times 18 \times \cos 75^\circ$$ 4. **Calculate each term:** $$20^2 = 400$$ $$18^2 = 324$$ $$2 \times 20 \times 18 = 720$$ 5. **Calculate $\cos 75^\circ$:** $$\cos 75^\circ \approx 0.2588$$ 6. **Plug in the cosine value:** $$c^2 = 400 + 324 - 720 \times 0.2588$$ 7. **Multiply:** $$720 \times 0.2588 = 186.336$$ 8. **Simplify:** $$c^2 = 400 + 324 - 186.336 = 724 - 186.336 = 537.664$$ 9. **Find $c$ by taking the square root:** $$c = \sqrt{537.664} \approx 23.19$$ **Final answer:** $$c \approx 23.19$$