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 the problem, we start with the known condition \( \cos \theta = \frac{-4}{5} \) and the range \( 0^{\circ} < \theta < 270^{\circ} \). This range indicates that \( \theta \) is in the second or third quadrant, where sine and cosine have specific signs. ### Step 1: Determine \( \sin \theta \) Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] we can substitute the value of \( \cos \theta \): \[ \sin^2 \theta + \left(\frac{-4}{5}\right)^2 = 1 \] Calculating \( \left(\frac{-4}{5}\right)^2 \): \[ \sin^2 \theta + \frac{16}{25} = 1 \] Subtracting \( \frac{16}{25} \) from both sides: \[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \] Taking the square root: \[ \sin \theta = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5} \] Since \( \theta \) is in the second quadrant, where sine is positive: \[ \sin \theta = \frac{3}{5} \] ### Step 2: Calculate \( 2.1 \sin \theta + \cos \theta \) Now we can calculate \( 2.1 \sin \theta + \cos \theta \): \[ 2.1 \sin \theta + \cos \theta = 2.1 \left(\frac{3}{5}\right) + \left(\frac{-4}{5}\right) \] Calculating \( 2.1 \left(\frac{3}{5}\right) \): \[ 2.1 \cdot \frac{3}{5} = \frac{6.3}{5} \] Now substituting back: \[ \frac{6.3}{5} - \frac{4}{5} = \frac{6.3 - 4}{5} = \frac{2.3}{5} \] ### Step 3: Calculate \( 22.3 \tan \theta + \sin \theta \) Next, we need to find \( \tan \theta \): \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{-4}{5}} = \frac{3}{-4} = -\frac{3}{4} \] Now we can calculate \( 22.3 \tan \theta + \sin \theta \): \[ 22.3 \tan \theta + \sin \theta = 22.3 \left(-\frac{3}{4}\right) + \frac{3}{5} \] Calculating \( 22.3 \left(-\frac{3}{4}\right) \): \[ 22.3 \cdot -\frac{3}{4} = -\frac{66.9}{4} = -16.725 \] Now substituting back: \[ -16.725 + \frac{3}{5} = -16.725 + 0.6 = -16.125 \] ### Step 4: Calculate \( 2.3 \frac{\sin \theta}{\cos \theta} \) Finally, we calculate \( 2.3 \frac{\sin \theta}{\cos \theta} \): \[ 2.3 \frac{\sin \theta}{\cos \theta} = 2.3 \tan \theta = 2.3 \left(-\frac{3}{4}\right) \] Calculating: \[ 2.3 \cdot -\frac{3}{4} = -\frac{6.9}{4} = -1.725 \] ### Summary of Results 1. \( 2.1 \sin \theta + \cos \theta = \frac{2.3}{5} \) 2. \( 22.3 \tan \theta + \sin \theta = -16.125 \) 3. \( 2.3 \frac{\sin \theta}{\cos \theta} = -1.725 \)