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

To find the requested values, we first need to determine the cosine of angle \( \alpha \) and the sine of angle \( \beta \). ### Step 1: Find \( \cos \alpha \) Given \( \sin \alpha = \frac{7}{25} \): \[ \cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \left(\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{576}{625} \] Thus, \[ \cos \alpha = \sqrt{\frac{576}{625}} = \frac{24}{25} \] ### Step 2: Find \( \sin \beta \) Given \( \cos \beta = \frac{8 \sqrt{89}}{89} \) (and since \( \beta \) is in the fourth quadrant, sine is negative): \[ \sin^2 \beta + \cos^2 \beta = 1 \implies \sin^2 \beta = 1 - \left(\frac{8\sqrt{89}}{89}\right)^2 \] Calculating \( \left(\frac{8\sqrt{89}}{89}\right)^2 \): \[ = \frac{64 \cdot 89}{7921} = \frac{5696}{7921} \] So, \[ \sin^2 \beta = 1 - \frac{5696}{7921} = \frac{7921 - 5696}{7921} = \frac{3225}{7921} \] Thus, \[ \sin \beta = -\sqrt{\frac{3225}{7921}} = -\frac{\sqrt{3225}}{89} \] We can simplify \( \sqrt{3225} \) as: \[ \sqrt{3225} = \sqrt{25 \cdot 129} = 5\sqrt{129} \] So, \[ \sin \beta = -\frac{5\sqrt{129}}{89} \] ### Step 3: Calculate \( \sin(\alpha + \beta) \) Using the sine addition formula: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \] Substituting in our known values: \[ \sin(\alpha + \beta) = \left(\frac{7}{25} \cdot \frac{8\sqrt{89}}{89}\right) + \left(\frac{24}{25} \cdot -\frac{5\sqrt{129}}{89}\right) \] Calculating each term: \[ = \frac{56\sqrt{89}}{2225} - \frac{120\sqrt{129}}{2225} \] Combining gives: \[ \sin(\alpha + \beta) = \frac{56\sqrt{89} - 120\sqrt{129}}{2225} \] ### Answers: \[ \text{(a) } \sin (\alpha + \beta) = \frac{56\sqrt{89} - 120\sqrt{129}}{2225} \] To find the remaining values: ### Step 4: Calculate \( \cos(\alpha + \beta) \) Using the cosine addition formula: \[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \] Substituting in known values: \[ \cos(\alpha + \beta) = \left(\frac{24}{25} \cdot \frac{8\sqrt{89}}{89}\right) - \left(\frac{7}{25} \cdot -\frac{5\sqrt{129}}{89}\right) \] Calculating: \[ = \frac{192\sqrt{89}}{2225} + \frac{35\sqrt{129}}{2225} \] So: \[ \cos(\alpha + \beta) = \frac{192\sqrt{89} + 35\sqrt{129}}{2225} \] ### Step 5: Calculate \( \sin(\alpha - \beta) \) Using the sine subtraction formula: \[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \] Substituting gives: \[ \sin(\alpha - \beta) = \left(\frac{7}{25} \cdot \frac{8\sqrt{89}}{89}\right) - \left