If \( \sin \theta=-\frac{3}{5} \) and \( \cos \beta=\frac{4}{5}, \cos \theta<0 \) and \( \sin \beta<0 \), determine the value of \( \sin \theta \cos \beta-\cos \theta \sin \beta \)

To solve for \( \sin \theta \cos \beta - \cos \theta \sin \beta \), we can use the sine subtraction formula: \[ \sin(\theta - \beta) = \sin \theta \cos \beta - \cos \theta \sin \beta \] We need to find \( \sin(\theta - \beta) \) by determining \( \sin \theta \), \( \cos \theta \), \( \sin \beta \), and \( \cos \beta \). ### Step 1: Determine \( \cos \theta \) Given \( \sin \theta = -\frac{3}{5} \) and \( \cos \theta < 0 \), we can use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the value of \( \sin \theta \): \[ \left(-\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \] Calculating \( \sin^2 \theta \): \[ \frac{9}{25} + \cos^2 \theta = 1 \] Now, solving for \( \cos^2 \theta \): \[ \cos^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \] Taking the square root: \[ \cos \theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \] ### Step 2: Determine \( \sin \beta \) Given \( \cos \beta = \frac{4}{5} \) and \( \sin \beta < 0 \), we again use the Pythagorean identity: \[ \sin^2 \beta + \cos^2 \beta = 1 \] Substituting the value of \( \cos \beta \): \[ \sin^2 \beta + \left(\frac{4}{5}\right)^2 = 1 \] Calculating \( \cos^2 \beta \): \[ \sin^2 \beta + \frac{16}{25} = 1 \] Now, solving for \( \sin^2 \beta \): \[ \sin^2 \beta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \] Taking the square root: \[ \sin \beta = -\sqrt{\frac{9}{25}} = -\frac{3}{5} \] ### Step 3: Calculate \( \sin \theta \cos \beta - \cos \theta \sin \beta \) Now we have: - \( \sin \theta = -\frac{3}{5} \) - \( \cos \theta = -\frac{4}{5} \) - \( \sin \beta = -\frac{3}{5} \) - \( \cos \beta = \frac{4}{5} \) Substituting these values into the expression: \[ \sin \theta \cos \beta - \cos \theta \sin \beta = \left(-\frac{3}{5}\right) \left(\frac{4}{5}\right) - \left(-\frac{4}{5}\right) \left(-\frac{3}{5}\right) \] Calculating each term: \[ = -\frac{12}{25} - \frac{12}{25} = -\frac{12}{25} - \frac{12}{25} = -\frac{24}{25} \] Thus, the final value is: \[ \boxed{-\frac{24}{25}} \]