Q 2. Find the values of \( \cos \theta \) and \( \tan \theta \), if \( \sin \theta=-\frac{3}{5} \) and \( \pi<\theta<\frac{3 \pi}{2} \)

To find the values of \( \cos \theta \) and \( \tan \theta \) given that \( \sin \theta = -\frac{3}{5} \) and \( \pi < \theta < \frac{3\pi}{2} \), we can follow these steps: 1. **Determine the quadrant**: Since \( \theta \) is in the interval \( \pi < \theta < \frac{3\pi}{2} \), it is in the third quadrant. In this quadrant, both sine and cosine are negative. 2. **Use the Pythagorean identity**: We know that: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting \( \sin \theta = -\frac{3}{5} \): \[ \left(-\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{9}{25} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = 1 - \frac{9}{25} \] \[ \cos^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \] Taking the square root, we find: \[ \cos \theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \] (We take the negative root because cosine is negative in the third quadrant.) 3. **Calculate \( \tan \theta \)**: The tangent function is defined as: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] Substituting the values we have: \[ \tan \theta = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4} \] Thus, the values are: \[ \cos \theta = -\frac{4}{5}, \quad \tan \theta = \frac{3}{4} \]