Use trigonometric ratios to find the size of the angle marked θ and β in each of the following right-angled triangles.

Here are the solutions for finding the angles $\theta$ and $\beta$ in each right-angled triangle.

2. Use trigonometric ratios to find the size of the angle marked $\theta$ and $\beta$ in each of the following right-angled triangles.

a) For the first triangle: Step 1: To find $\theta$, use the tangent ratio, as the opposite and adjacent sides are given. $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ $\tan(\theta) = \frac{5 \text{ cm}}{8 \text{ cm}}$ $\theta = \arctan\left(\frac{5}{8}\right)$ $\theta \approx 32.0^\circ$ Step 2: To find $\beta$, use the property that the sum of the acute angles in a right-angled triangle is $90^\circ$. $\beta = 90^\circ - \theta$ $\beta = 90^\circ - 32.0^\circ$ $\beta \approx 58.0^\circ$ The angles are: $\boxed{\theta \approx 32.0^\circ, \beta \approx 58.0^\circ}$

b) For the second triangle: Step 1: To find $\theta$, use the sine ratio, as the opposite side and hypotenuse are given. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin(\theta) = \frac{4 \text{ cm}}{10 \text{ cm}}$ $\sin(\theta) = 0.4$ $\theta = \arcsin(0.4)$ $\theta \approx 23.6^\circ$ Step 2: To find $\beta$, use the property that the sum of the acute angles in a right-angled triangle is $90^\circ$. $\beta = 90^\circ - \theta$ $\beta = 90^\circ - 23.6^\circ$ $\beta \approx 66.4^\circ$ The angles are: $\boxed{\theta \approx 23.6^\circ, \beta \approx 66.4^\circ}$

c) For the third triangle: Step 1: To find $\theta$, use the sine ratio, as the opposite side and hypotenuse are given. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin(\theta) = \frac{16 \text{ cm}}{25 \text{ cm}}$ $\sin(\theta) = 0.64$ $\theta = \arcsin(0.64)$ $\theta \approx 39.8^\circ$ Step 2: To find $\beta$, use the property that the sum of the acute angles in a right-angled triangle is $90^\circ$. $\beta = 90^\circ - \theta$ $\beta = 90^\circ - 39.8^\circ$ $\beta \approx 50.2^\circ$ The angles are: $\boxed{\theta \approx 39.8^\circ, \beta \approx 50.2^\circ}$