Find the exact value of the indicated trigonometric function of \( \theta \). \[ \text { 16) } \tan \theta=\frac{24}{7}, 180^{\circ}<\theta<270^{\circ} \quad \text { Find } \cos \theta \]

To find \(\cos \theta\) when \(\tan \theta = \frac{24}{7}\) and \(180^{\circ}<\theta<270^{\circ}\), we start by recalling the identity relating \(\tan\) to \(\sin\) and \(\cos\): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{24}{7}. \] In the third quadrant, both sine and cosine are negative, so we can let: \[ \sin \theta = -24k \quad \text{and} \quad \cos \theta = -7k \] for some positive \(k\). Now, using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1, \] we substitute: \[ (-24k)^2 + (-7k)^2 = 1. \] This simplifies to: \[ 576k^2 + 49k^2 = 1 \quad \Rightarrow \quad 625k^2 = 1 \quad \Rightarrow \quad k^2 = \frac{1}{625} \quad \Rightarrow \quad k = \frac{1}{25}. \] Now, we can find \(\sin \theta\) and \(\cos \theta\): \[ \sin \theta = -24 \left(\frac{1}{25}\right) = -\frac{24}{25}, \] \[ \cos \theta = -7 \left(\frac{1}{25}\right) = -\frac{7}{25}. \] Therefore, the exact value of \(\cos \theta\) is: \[ \cos \theta = -\frac{7}{25}. \]