1. **Stating the problem:** We need to find the roof angle (takvinkel) in two right triangles given the lengths of the opposite and adjacent sides. 2. **Formula used:** The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ To find the angle $\theta$, we use the inverse tangent (arctan): $$\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$$ 3. **Important rule:** The angle is measured in degrees, so ensure your calculator is set to degrees. --- ### a) Triangle with opposite = 4 and adjacent = 6 4. Calculate the ratio: $$\tan(\theta) = \frac{4}{6}$$ 5. Simplify the fraction: $$\tan(\theta) = \frac{\cancel{2} \times 2}{\cancel{2} \times 3} = \frac{2}{3}$$ 6. Find the angle: $$\theta = \tan^{-1}\left(\frac{2}{3}\right) \approx 33.69^\circ$$ --- ### b) Triangle with opposite = 2 and adjacent = 5 7. Calculate the ratio: $$\tan(\theta) = \frac{2}{5}$$ 8. This fraction is already simplified. 9. Find the angle: $$\theta = \tan^{-1}\left(\frac{2}{5}\right) \approx 21.80^\circ$$ --- **Final answers:** - a) Takvinkel $\approx 33.69^\circ$ - b) Takvinkel $\approx 21.80^\circ$