Find the exact value of each of the following under the given conditions. \[ \sin \alpha=\frac{7}{25}, 0<\alpha<\frac{\pi}{2} ; \cos \beta=\frac{8 \sqrt{89}}{89},-\frac{\pi}{2}<\beta<0 \] \( \begin{array}{llll}\text { (a) } \sin (\alpha+\beta) & \text { (b) } \cos (\alpha+\beta) & \text { (c) } \sin (\alpha-\beta) & \text { (d) } \tan (\alpha-\beta)\end{array} \)

To find the exact values requested, we will first need to calculate the missing trigonometric values based on the given information. We know that: 1. \(\sin \alpha = \frac{7}{25}\) 2. In quadrant I, we can use the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\) to find \(\cos \alpha\): \[ \cos^2 \alpha = 1 - \sin^2 \alpha = 1 - \left(\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{576}{625} \] \[ \cos \alpha = \sqrt{\frac{576}{625}} = \frac{24}{25} \] 3. For \(\beta\), we know that \(\cos \beta = \frac{8 \sqrt{89}}{89}\) and \(\beta\) is in quadrant IV, where \(\sin \beta\) can be found using the identity \(\sin^2 \beta + \cos^2 \beta = 1\): \[ \sin^2 \beta = 1 - \cos^2 \beta = 1 - \left( \frac{8 \sqrt{89}}{89} \right)^2 = 1 - \frac{64 \cdot 89}{89^2} = \frac{89^2 - 64 \cdot 89}{89^2} = \frac{89(89 - 64)}{89^2} = \frac{89 \cdot 25}{89^2} = \frac{25}{89} \] Since \(\beta\) is in the fourth quadrant: \[ \sin \beta = -\sqrt{\frac{25}{89}} = -\frac{5}{\sqrt{89}} \] Now we have: - \(\sin \alpha = \frac{7}{25}\) - \(\cos \alpha = \frac{24}{25}\) - \(\sin \beta = -\frac{5}{\sqrt{89}}\) - \(\cos \beta = \frac{8 \sqrt{89}}{89}\) Now, we can compute the required values using the sine and cosine sum and difference formulas. **(a) \(\sin(\alpha + \beta)\)**: \[ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \] \[ = \left(\frac{7}{25}\right) \left(\frac{8 \sqrt{89}}{89}\right) + \left(\frac{24}{25}\right) \left(-\frac{5}{\sqrt{89}}\right) \] \[ = \frac{56 \sqrt{89}}{2225} - \frac{120}{25 \sqrt{89}} = \frac{56 \sqrt{89}}{2225} - \frac{480}{2225} = \frac{56 \sqrt{89} - 480}{2225} \] **(b) \(\cos(\alpha + \beta)\)**: \[ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \] \[ = \left(\frac{24}{25}\right) \left(\frac{8 \sqrt{89}}{89}\right) - \left(\frac{7}{25}\right) \left(-\frac{5}{\sqrt{89}}\right) \] \[ = \frac{192 \sqrt{89}}{2225} + \frac{35}{25 \sqrt{89}} = \frac{192 \sqrt{89}}{2225} + \frac{35 \sqrt{89}}{2225} = \frac{(192 + 35)\sqrt{89}}{2225} = \frac{227 \sqrt{89}}{2225} \] **(c) \(\sin(\alpha - \beta)\)**: \[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \] \[ = \left(\frac{7}{25}\right) \left(\frac{8 \sqrt{89}}{89}\right) - \left(\frac{24}{25}\right) \left(-\frac{5}{\sqrt{89}}\right) \] \[ = \frac{56 \sqrt{89}}{2225