2. Given the following equations: a) Determine the general solution of each of the following equations. Give your answers correct to the nearest degree. \( \begin{array}{ll}\text { b) List the solutions in the interval }\left[-180^{\circ} ; 180^{\circ}\right] \\ \text { (i) } 2 \sin \theta=0,64 & \text { (ii) } \sin 2 \theta=0,64 \\ \text { (iii) } 3 \tan \theta=-1,29 & \text { (iv) } \tan 3 \theta=-1,29 \\ \text { (v) } 3+\tan \theta=-1,29 & \text { (vi) } 3-3 \cos 2 \theta=2,28 \\ \text { (vii) } \sin \theta=0 & \text { (viii) } \cos 2 \theta=-\frac{1}{2} \\ \text { (ix) } 2 \tan \theta=-1 & \text { (xii) } \sin \frac{\theta}{2}=-\frac{\sqrt{3}}{2} \\ \text { (xi) } \cos 3 \theta=1 & \end{array} \)

To solve the given equations, we will follow these steps: 1. Solve each equation for the general solution. 2. List the specific solutions within the interval \([-180^\circ, 180^\circ]\). Let's go through each equation step by step. ### (i) \(2 \sin \theta = 0.64\) **Step 1: Solve for \(\sin \theta\)** \[ \sin \theta = \frac{0.64}{2} = 0.32 \] **Step 2: Find the general solution** The general solution for \(\sin \theta = k\) is given by: \[ \theta = \arcsin(k) + 360^\circ n \quad \text{or} \quad \theta = 180^\circ - \arcsin(k) + 360^\circ n \] **Step 3: Calculate \(\theta\)** \[ \theta = \arcsin(0.32) \approx 18.9^\circ \] Thus, the general solutions are: \[ \theta = 18.9^\circ + 360^\circ n \quad \text{and} \quad \theta = 180^\circ - 18.9^\circ + 360^\circ n \] \[ \theta = 18.9^\circ + 360^\circ n \quad \text{and} \quad \theta = 161.1^\circ + 360^\circ n \] **Step 4: List solutions in \([-180^\circ, 180^\circ]\)** For \(n = 0\): - \(18.9^\circ\) - \(161.1^\circ\) (not in the interval) For \(n = -1\): - \(18.9^\circ - 360^\circ = -341.1^\circ\) (not in the interval) - \(161.1^\circ - 360^\circ = -198.9^\circ\) (not in the interval) Thus, the only solution in the interval is: \[ \theta \approx 18.9^\circ \] ### (ii) \(\sin 2\theta = 0.64\) **Step 1: Solve for \(2\theta\)** \[ 2\theta = \arcsin(0.64) + 360^\circ n \quad \text{or} \quad 2\theta = 180^\circ - \arcsin(0.64) + 360^\circ n \] **Step 2: Calculate \(2\theta\)** \[ 2\theta = 39.8^\circ + 360^\circ n \quad \text{and} \quad 2\theta = 140.2^\circ + 360^\circ n \] **Step 3: Solve for \(\theta\)** \[ \theta = 19.9^\circ + 180^\circ n \quad \text{and} \quad \theta = 70.1^\circ + 180^\circ n \] **Step 4: List solutions in \([-180^\circ, 180^\circ]\)** For \(n = 0\): - \(19.9^\circ\) - \(70.1^\circ\) For \(n = -1\): - \(19.9^\circ - 180^\circ = -160.1^\circ\) - \(70.1^\circ - 180^\circ = -109.9^\circ\) Thus, the solutions in the interval are: \[ \theta \approx 19.9^\circ, 70.1^\circ, -160.1^\circ, -109.9^\circ \] ### (iii) \(3 \tan \theta = -1.29\) **Step 1: Solve for \(\tan \theta\)** \[ \tan \theta = \frac{-1.29}{3} \approx -0.43 \] **Step 2: Find the general solution** The general solution for \(\tan \theta = k\) is given by: \[ \theta = \arctan(k) + 180^\circ n \] **Step 3: Calculate \(\theta\)** \[ \theta = \arctan(-0.43) \approx -23.5^\circ \] Thus, the general solutions are: \[ \theta = -23.5^\circ + 180^\circ n \] **Step 4: List solutions in \([-180^\circ, 180^\circ]\)** For \(n = 0\): - \(-23.5^\circ\) For \(n = -1\): - \(-23.5^\circ - 180^\circ = -203.5^\circ\) (not in the interval) Thus, the only solution in the interval is: \[ \theta \approx -23.5^\circ \] ### (iv) \(\tan 3\theta = -1.29\) **Step 1: Solve for \(3\theta\)** \[ 3\theta = \arctan(-1.29) + 180^\circ n \] **Step 2: Calculate \(3\theta\)** \[ 3\theta \approx -52.5^\circ + 180^\circ n \] **Step 3: Solve for \(\theta\)** \[ \theta \approx -17.5^\circ + 60^\circ n \] **Step 4: List solutions in \([-180^\circ, 180^\circ]\)** For \(n = 0\): - \(-17.5^\circ\) For \(n = -1\): - \(-17.5^\circ - 60^\circ = -77.5^\circ\) For \(n = 1\): - \(-17.5^\circ + 60^\circ = 42.5^\circ\) Thus, the solutions in the interval are: \[ \theta \approx -17.5^\circ, -77.5^\circ, 42.5^\circ \] ### (v) \(3 + \tan \theta = -1.29\) **Step 1: Solve for \(\tan \theta\)** \[ \tan \theta = -1.29 - 3 = -4.29 \] **Step 2: Find the general solution** \[ \theta = \arctan(-4.29) + 180^\circ n \] **Step 3: Calculate \(\theta\)** \[ \theta \approx -77.5^\circ + 180^\circ n \] **Step 4: List solutions in \([-180^\circ, 180^\circ]\)** For \(n = 0\): - \(-77.5^\circ\