1. Problem: Find the length $x$ in each right-angled triangle given an angle and a side length. 2. Understand that in right-angled triangles, we can use trigonometric ratios (sine, cosine, tangent) to find missing sides. **(a)** Given angle $30^\circ$ and adjacent side length 6 cm, find opposite side $x$. - Use tangent because $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. - So, $\tan(30^\circ) = \frac{x}{6}$. - Calculate $\tan(30^\circ) = \frac{\sqrt{3}}{3}$. - Solve for $x$: $$x = 6 \times \tan(30^\circ) = 6 \times \frac{\sqrt{3}}{3} = 2\sqrt{3} \approx 3.464.$$ cm **(b)** Given angle $42^\circ$ and opposite side length 8 cm, find adjacent side $x$. - Use tangent: $\tan(42^\circ) = \frac{8}{x}$. - So, $x = \frac{8}{\tan(42^\circ)}$. - Calculate $\tan(42^\circ) \approx 0.9004$. - Thus, $x \approx \frac{8}{0.9004} \approx 8.887$ cm. **(c)** Without specific angle or side length, we cannot determine $x$. More information is needed. Final answers: - (a) $x = 2\sqrt{3} \approx 3.464$ cm - (b) $x \approx 8.887$ cm - (c) Insufficient data to solve.