1. **State the problem:** We are given a triangle with sides $a=20$, $b=18$, and angle $C=15^\circ$. We need to 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 the known values:** $$c^2 = 20^2 + 18^2 - 2 \times 20 \times 18 \times \cos 15^\circ$$ 4. **Calculate each term:** $$20^2 = 400$$ $$18^2 = 324$$ $$2 \times 20 \times 18 = 720$$ 5. **Calculate $\cos 15^\circ$:** $$\cos 15^\circ \approx 0.9659$$ 6. **Plug in the cosine value:** $$c^2 = 400 + 324 - 720 \times 0.9659$$ 7. **Multiply:** $$720 \times 0.9659 = 695.448$$ 8. **Simplify:** $$c^2 = 724 - 695.448 = 28.552$$ 9. **Find $c$ by taking the square root:** $$c = \sqrt{28.552} \approx 5.34$$ **Final answer:** $$\boxed{c \approx 5.34}$$