1. **Problem Statement:** Find the missing side length $x$ and the missing angle in a right triangle where side $AC=12$ cm, angle $A=25^\circ$, and side $BC=x$ is unknown. 2. **Formula and Rules:** In a right triangle, the sides relate to angles by sine, cosine, and tangent functions: - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Identify sides relative to angle $A$:** - Opposite side to angle $A$ is $BC = x$ - Adjacent side to angle $A$ is $AC = 12$ cm - Hypotenuse is $AB$ 4. **Find the missing angle $B$:** Since the triangle is right angled at $C$, angle $C=90^\circ$. Sum of angles in triangle is $180^\circ$: $$ B = 180^\circ - 90^\circ - 25^\circ = 65^\circ $$ 5. **Find hypotenuse $AB$ using cosine:** $$ \cos(25^\circ) = \frac{AC}{AB} = \frac{12}{AB} $$ Rearranged: $$ AB = \frac{12}{\cos(25^\circ)} $$ Calculate: $$ AB \approx \frac{12}{0.9063} \approx 13.24\text{ cm} $$ 6. **Find missing side $x = BC$ using sine:** $$ \sin(25^\circ) = \frac{BC}{AB} = \frac{x}{13.24} $$ Rearranged: $$ x = 13.24 \times \sin(25^\circ) $$ Calculate: $$ x \approx 13.24 \times 0.4226 \approx 5.59\text{ cm} $$ **Final answers:** - Missing side $x = BC \approx 5.59$ cm - Missing angle $B = 65^\circ$