1. **Problem statement:** We have six right-angled triangles labeled (a) through (f). Each triangle has one angle (other than the right angle), one known side, and one unknown side $x$. We need to find $x$ in each case. 2. **Formula and rules:** In a right triangle, the sides relate to angles via sine, cosine, and tangent: - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ We identify which side is opposite, adjacent, or hypotenuse relative to the given angle and use the appropriate ratio. --- ### (a) Given hypotenuse = 10 cm, angle = 35°, find adjacent side $x$. - Use cosine: $\cos(35^\circ) = \frac{x}{10}$ - Solve for $x$: $x = 10 \times \cos(35^\circ)$ - Calculate: $x \approx 10 \times 0.8192 = 8.192$ cm --- ### (b) Given adjacent side = 8 cm, angle = 40°, find opposite side $x$. - Use tangent: $\tan(40^\circ) = \frac{x}{8}$ - Solve for $x$: $x = 8 \times \tan(40^\circ)$ - Calculate: $x \approx 8 \times 0.8391 = 6.713$ cm --- ### (c) Given opposite side = 6 cm, angle = 70°, find adjacent side $x$. - Use tangent: $\tan(70^\circ) = \frac{6}{x}$ - Solve for $x$: $x = \frac{6}{\tan(70^\circ)}$ - Calculate: $x \approx \frac{6}{2.747} = 2.183$ cm --- ### (d) Given opposite side = 18 cm, angle = 22°, find adjacent side $x$. - Use tangent: $\tan(22^\circ) = \frac{18}{x}$ - Solve for $x$: $x = \frac{18}{\tan(22^\circ)}$ - Calculate: $x \approx \frac{18}{0.4040} = 44.554$ cm --- ### (e) Given opposite side = 2.8 cm, angle = 31°, find adjacent side $x$. - Use tangent: $\tan(31^\circ) = \frac{2.8}{x}$ - Solve for $x$: $x = \frac{2.8}{\tan(31^\circ)}$ - Calculate: $x \approx \frac{2.8}{0.6009} = 4.659$ cm --- ### (f) Given opposite side = 75 cm, angle = 55°, find adjacent side $x$. - Use tangent: $\tan(55^\circ) = \frac{75}{x}$ - Solve for $x$: $x = \frac{75}{\tan(55^\circ)}$ - Calculate: $x \approx \frac{75}{1.4281} = 52.54$ cm --- **Final answers:** - (a) $x \approx 8.19$ cm - (b) $x \approx 6.71$ cm - (c) $x \approx 2.18$ cm - (d) $x \approx 44.55$ cm - (e) $x \approx 4.66$ cm - (f) $x \approx 52.54$ cm These results use standard trigonometric ratios and approximate decimal values for sine, cosine, and tangent functions.