1. **Problem (a):** Calculate the length of the side $w$, opposite the $15^\circ$ angle, in a right triangle with hypotenuse $10$ cm. 2. Use the sine function because sine relates the opposite side and hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ 3. Substitute values: $$\sin(15^\circ) = \frac{w}{10}$$ 4. Solve for $w$: $$w = 10 \times \sin(15^\circ)$$ 5. Compute $\sin(15^\circ) \approx 0.2588$, so: $$w \approx 10 \times 0.2588 = 2.588$$ cm --- 6. **Problem (b):** Calculate the length of side $x$, opposite the $19^\circ$ angle, in a right triangle with hypotenuse $12$ cm. 7. Again, apply the sine function: $$\sin(19^\circ) = \frac{x}{12}$$ 8. Solve for $x$: $$x = 12 \times \sin(19^\circ)$$ 9. Calculate $\sin(19^\circ) \approx 0.3256$, so: $$x \approx 12 \times 0.3256 = 3.907$$ cm **Final answers:** - $w \approx 2.59$ cm - $x \approx 3.91$ cm