1. **State the problem:** Find the area of a triangle with sides $b=15$ cm, $c=24$ cm, and angle $\angle A=103^\circ$ between them. 2. **Formula used:** The area $A$ of a triangle given two sides and the included angle is $$A = \frac{1}{2}bc\sin(\angle A)$$ This formula comes from the fact that the height can be expressed using the sine of the included angle. 3. **Calculate the sine of the angle:** $$\sin(103^\circ) \approx 0.9744$$ 4. **Plug in the values:** $$A = \frac{1}{2} \times 15 \times 24 \times 0.9744$$ 5. **Simplify step-by-step:** $$A = \frac{1}{2} \times 360 \times 0.9744$$ $$A = 180 \times 0.9744$$ 6. **Final calculation:** $$A \approx 175.4 \text{ cm}^2$$ **Answer:** The area of the triangle is approximately $175.4$ square centimeters.