2) Find the exact value of \( \cos (a-\beta) \) if \( \sin \alpha=-\frac{3}{5}, \frac{3 \pi}{2}<\alpha<2 \pi \) and \( \cos \beta=-\frac{5}{13}, \frac{\pi}{2}<\beta<\pi \). a) \( 0 \frac{12}{13} \) b) \( 0-\frac{56}{65} \) c) \( 0 \frac{56}{65} \) d) \( 0-\frac{12}{13} \)

To find the exact value of \( \cos(a - \beta) \), we can use the cosine subtraction formula: \[ \cos(a - \beta) = \cos a \cos \beta + \sin a \sin \beta \] First, we need to determine \( \cos a \) and \( \sin \beta \). ### Step 1: Find \( \cos a \) Given \( \sin \alpha = -\frac{3}{5} \) and \( \frac{3\pi}{2} < \alpha < 2\pi \), we are in the fourth quadrant where cosine is positive. We can use the Pythagorean identity: \[ \sin^2 \alpha + \cos^2 \alpha = 1 \] Substituting the value of \( \sin \alpha \): \[ \left(-\frac{3}{5}\right)^2 + \cos^2 \alpha = 1 \] \[ \frac{9}{25} + \cos^2 \alpha = 1 \] \[ \cos^2 \alpha = 1 - \frac{9}{25} = \frac{16}{25} \] \[ \cos \alpha = \sqrt{\frac{16}{25}} = \frac{4}{5} \] Since \( \alpha \) is in the fourth quadrant, \( \cos \alpha = \frac{4}{5} \). ### Step 2: Find \( \sin \beta \) Given \( \cos \beta = -\frac{5}{13} \) and \( \frac{\pi}{2} < \beta < \pi \), we are in the second quadrant where sine is positive. Again, we use the Pythagorean identity: \[ \sin^2 \beta + \cos^2 \beta = 1 \] Substituting the value of \( \cos \beta \): \[ \sin^2 \beta + \left(-\frac{5}{13}\right)^2 = 1 \] \[ \sin^2 \beta + \frac{25}{169} = 1 \] \[ \sin^2 \beta = 1 - \frac{25}{169} = \frac{144}{169} \] \[ \sin \beta = \sqrt{\frac{144}{169}} = \frac{12}{13} \] Since \( \beta \) is in the second quadrant, \( \sin \beta = \frac{12}{13} \). ### Step 3: Calculate \( \cos(a - \beta) \) Now we can substitute \( \cos a \), \( \sin a \), \( \cos \beta \), and \( \sin \beta \) into the cosine subtraction formula: \[ \cos(a - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \] \[ = \left(\frac{4}{5}\right)\left(-\frac{5}{13}\right) + \left(-\frac{3}{5}\right)\left(\frac{12}{13}\right) \] \[ = -\frac{20}{65} - \frac{36}{65} \] \[ = -\frac{20 + 36}{65} = -\frac{56}{65} \] Thus, the exact value of \( \cos(a - \beta) \) is \[ \boxed{0 - \frac{56}{65}} \] This corresponds to option (b).