\( \cos \theta=\frac{-24}{25} \) and \( 180^{\circ} \leqslant \theta \leqslant 270^{\circ} \) value of \( \theta \)

To find the angle \( \theta \) given that \( \cos \theta = \frac{-24}{25} \) and \( 180^{\circ} \leqslant \theta \leqslant 270^{\circ} \), we can follow these steps: 1. **Determine the reference angle**: Since \( \cos \theta \) is negative, \( \theta \) is in the third quadrant (which is confirmed by the given range). The reference angle \( \theta_r \) can be found using the cosine inverse function: \[ \theta_r = \cos^{-1}\left(-\frac{24}{25}\right) \] However, since we are looking for the angle in the third quadrant, we can find \( \theta \) directly. 2. **Calculate the angle in the third quadrant**: The angle in the third quadrant can be calculated as: \[ \theta = 180^{\circ} + \theta_r \] To find \( \theta_r \), we first calculate \( \cos^{-1}\left(\frac{24}{25}\right) \): \[ \theta_r = \cos^{-1}\left(\frac{24}{25}\right) \] Using a calculator, we find: \[ \theta_r \approx 25.84^{\circ} \] 3. **Find the angle in the third quadrant**: \[ \theta = 180^{\circ} + 25.84^{\circ} \approx 205.84^{\circ} \] Thus, the value of \( \theta \) is approximately: \[ \theta \approx 205.84^{\circ} \] This is the angle in the specified range where \( \cos \theta = \frac{-24}{25} \).