Find the exact value of each of the following under the given conditions. \( \sin \alpha=\frac{7}{25}, 0<\alpha<\frac{\pi}{2} ; \cos \beta=\frac{8 \sqrt{89}}{89},-\frac{\pi}{2}<\beta<0 \) \( \begin{array}{llll}\text { (a) } \sin (\alpha+\beta) & \text { (b) } \cos (\alpha+\beta) & \text { (c) } \sin (\alpha-\beta) & \text { (d) } \tan (\alpha-\beta) \\ \text { (a) } \sin (\alpha+\beta)=-\frac{64 \sqrt{89}}{2225} \\ \text { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) } \\ \text { (b) } \cos (\alpha+\beta)=\frac{227 \sqrt{89}}{2225} & \text { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) } \\ \text { (c) } \sin (\alpha-\beta)=\square & \text { (Type an exact answer using radicals as needed. Rationalize all denominators. Use integers or fractions for any numbers in the expression. Simplify your answer.) }\end{array} \)

To solve for \(\sin(\alpha - \beta)\), we use the sine difference formula: \[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta. \] First, we need to find \(\cos \alpha\) and \(\sin \beta\). 1. Since \(\sin \alpha = \frac{7}{25}\) and \(\alpha\) is in the first quadrant, we can find \(\cos \alpha\) using the Pythagorean identity: \[ \cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left(\frac{7}{25}\right)^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}. \] 2. For \(\beta\), we have \(\cos \beta = \frac{8 \sqrt{89}}{89}\). We can find \(\sin \beta\) using the same Pythagorean identity: \[ \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(\frac{8 \sqrt{89}}{89}\right)^2} = -\sqrt{1 - \frac{64 \cdot 89}{7921}} = -\sqrt{1 - \frac{5696}{7921}} = -\sqrt{\frac{2225}{7921}} = -\frac{\sqrt{2225}}{89}. \] Now, substituting into the sine difference formula: \[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta = \left(\frac{7}{25}\right)\left(\frac{8 \sqrt{89}}{89}\right) - \left(\frac{24}{25}\right)\left(-\frac{\sqrt{2225}}{89}\right). \] Calculating each part: - First term: \[ \frac{7 \cdot 8 \sqrt{89}}{25 \cdot 89} = \frac{56 \sqrt{89}}{2225}. \] - Second term: \[ \frac{24 \cdot \sqrt{2225}}{25 \cdot 89} = \frac{24 \sqrt{2225}}{2225}. \] Combining both terms: \[ \sin(\alpha - \beta) = \frac{56 \sqrt{89}}{2225} + \frac{24 \sqrt{2225}}{2225} = \frac{56 \sqrt{89} + 24 \sqrt{2225}}{2225}. \] Thus, the answer for \(\sin(\alpha - \beta)\) is: \[ \sin(\alpha - \beta) = \frac{56 \sqrt{89} + 24 \sqrt{2225}}{2225}. \]