Cos Alpha Beta F1179E

1. **State the problem:** Find the exact value of $$\cos(\alpha - \beta)$$ given $$\sin \alpha = \frac{7}{25}$$ with $$\frac{\pi}{2} < \alpha < \pi$$ and $$\cos \beta = \frac{2}{5}$$ with $$0 < \beta < \frac{\pi}{2}$$. 2. **Recall the formula:** $$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$ 3. **Find $$\cos \alpha$$:** Since $$\sin^2 \alpha + \cos^2 \alpha = 1$$, $$\cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \left(\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{625 - 49}{625} = \frac{576}{625}$$ Because $$\frac{\pi}{2} < \alpha < \pi$$, $$\cos \alpha$$ is negative: $$\cos \alpha = -\frac{24}{25}$$ 4. **Find $$\sin \beta$$:** Similarly, $$\sin^2 \beta = 1 - \cos^2 \beta = 1 - \left(\frac{2}{5}\right)^2 = 1 - \frac{4}{25} = \frac{21}{25}$$ Since $$0 < \beta < \frac{\pi}{2}$$, $$\sin \beta$$ is positive: $$\sin \beta = \frac{\sqrt{21}}{5}$$ 5. **Substitute values into the formula:** $$\cos(\alpha - \beta) = \left(-\frac{24}{25}\right) \cdot \frac{2}{5} + \frac{7}{25} \cdot \frac{\sqrt{21}}{5} = -\frac{48}{125} + \frac{7\sqrt{21}}{125} = \frac{-48 + 7\sqrt{21}}{125}$$ 6. **Final answer:** $$\boxed{\frac{-48 + 7\sqrt{21}}{125}}$$