Q2 If \( \cos \theta=\frac{-4}{5} \) and \( 0^{\circ}<\theta<270 \) Determine \( 2.1 \sin \theta+\cos \theta \) \( 22.3 \tan \theta+\sin \theta \) \( 2.3 \frac{\sin \theta}{\cos \theta} \)

To solve for \( \sin \theta \) given \( \cos \theta = -\frac{4}{5} \) and \( 0^{\circ} < \theta < 270^{\circ} \), we first identify the quadrant. Since \( \cos \theta \) is negative and \( \theta \) is less than \( 270^{\circ} \), this means \(\theta\) lies in the second or third quadrant. Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \): 1. We find \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \] Thus, \( \sin \theta = \pm\sqrt{\frac{9}{25}} = \pm\frac{3}{5} \). 2. In the second quadrant, \( \sin \theta\) is positive: \(\sin \theta = \frac{3}{5}\). Now, we can evaluate the requested expressions: **For \(2.1 \sin \theta + \cos \theta\)**: \[ 2.1 \sin \theta + \cos \theta = 2.1\left(\frac{3}{5}\right) + \left(-\frac{4}{5}\right) = \frac{6.3}{5} - \frac{4}{5} = \frac{2.3}{5}. \] **For \(22.3 \tan \theta + \sin \theta\)**, we find \( \tan \theta\): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}. \] Then, \[ 22.3 \tan \theta + \sin \theta = 22.3\left(-\frac{3}{4}\right) + \frac{3}{5} = -16.725 + 0.6 = -16.125. \] **For \(2.3 \frac{\sin \theta}{\cos \theta}\)**: \[ 2.3 \frac{\sin \theta}{\cos \theta} = 2.3 \tan \theta = 2.3\left(-\frac{3}{4}\right) = -1.725. \] So, our final results are: - \( 2.1 \sin \theta + \cos \theta = \frac{2.3}{5} \) - \( 22.3 \tan \theta + \sin \theta = -16.125 \) - \( 2.3 \frac{\sin \theta}{\cos \theta} = -1.725 \)