1. **Problem statement:** Calculate the angle between vectors $\mathbf{a} = (1,2)$ and $\mathbf{c} = (-2,5)$ in degrees. 2. **Formula:** The angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$ is given by $$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}$$ where $\mathbf{u} \cdot \mathbf{v}$ is the dot product and $|\mathbf{u}|$, $|\mathbf{v}|$ are the magnitudes. 3. **Calculate the dot product:** $$\mathbf{a} \cdot \mathbf{c} = (1)(-2) + (2)(5) = -2 + 10 = 8$$ 4. **Calculate the magnitudes:** $$|\mathbf{a}| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$$ $$|\mathbf{c}| = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29}$$ 5. **Calculate cosine of the angle:** $$\cos \theta = \frac{8}{\sqrt{5} \times \sqrt{29}} = \frac{8}{\sqrt{145}}$$ 6. **Calculate the angle in radians:** $$\theta = \arccos\left(\frac{8}{\sqrt{145}}\right)$$ 7. **Convert to degrees:** $$\theta_{\text{degrees}} = \theta \times \frac{180}{\pi}$$ 8. **Numerical evaluation:** $$\sqrt{145} \approx 12.0416$$ $$\cos \theta \approx \frac{8}{12.0416} \approx 0.6644$$ $$\theta \approx \arccos(0.6644) \approx 0.8411 \text{ radians}$$ $$\theta_{\text{degrees}} \approx 0.8411 \times 57.2958 = 48.2^\circ$$ **Final answer:** The angle between vectors $\mathbf{a}$ and $\mathbf{c}$ is approximately $48.2^\circ$.