1. **Problem Statement:** Calculate the unknown side length $x$ in each right-angled triangle given the angles and sides, correct to 4 significant figures. 2. **Key formulas and rules:** - In a right-angled triangle, the sides relate to angles via sine, cosine, and tangent functions. - For angle $\theta$, hypotenuse $h$, opposite side $o$, adjacent side $a$: - $\sin(\theta) = \frac{o}{h}$ - $\cos(\theta) = \frac{a}{h}$ - $\tan(\theta) = \frac{o}{a}$ --- ### (a) Given: - Angle $70^\circ$ (bottom-right corner) - Adjacent side to $70^\circ$ is 13.0 - Hypotenuse = $x$ Using cosine: $$\cos(70^\circ) = \frac{13.0}{x} \implies x = \frac{13.0}{\cos(70^\circ)}$$ Calculate: $$\cos(70^\circ) \approx 0.3420$$ $$x = \frac{13.0}{0.3420} \approx 38.01$$ --- ### (b) Given: - Angle $22^\circ$ (bottom-left corner) - Opposite side to $22^\circ$ is 15.0 - Hypotenuse = $x$ Using sine: $$\sin(22^\circ) = \frac{15.0}{x} \implies x = \frac{15.0}{\sin(22^\circ)}$$ Calculate: $$\sin(22^\circ) \approx 0.3746$$ $$x = \frac{15.0}{0.3746} \approx 40.05$$ --- ### (c) Given: - Angle $23^\circ$ (top-left corner) - Adjacent side to $23^\circ$ is 17.0 - Hypotenuse = $x$ Using cosine: $$\cos(23^\circ) = \frac{17.0}{x} \implies x = \frac{17.0}{\cos(23^\circ)}$$ Calculate: $$\cos(23^\circ) \approx 0.9205$$ $$x = \frac{17.0}{0.9205} \approx 18.46$$ --- **Final answers (4 significant figures):** - (a) $x \approx 38.01$ - (b) $x \approx 40.05$ - (c) $x \approx 18.46$