7. Find the general solution of each of the following equations \( \begin{array}{ll}\text { (a) } \cos 2 \theta=0,4 & \text { (b) } 5 \sin \theta+14 \cos \theta=0\end{array} \) 8. Solve for \( x \) if \( \sin \left(x+30^{\circ}\right)=-0.2 \) where \( x \in\left(-180^{\circ} ; 0^{\circ}\right) \)

Let's solve each of the problems step-by-step. --- ### **Problem 7** **Find the general solution of each of the following equations:** **(a) \( \cos 2\theta = 0.4 \)** **Solution:** To find the general solution for \( \cos 2\theta = 0.4 \), follow these steps: 1. **Find the Principal Solutions:** \[ 2\theta = \cos^{-1}(0.4) + 2\pi n \quad \text{and} \quad 2\theta = -\cos^{-1}(0.4) + 2\pi n \] where \( n \) is any integer. 2. **Solve for \( \theta \):** Divide both sides by 2: \[ \theta = \frac{1}{2} \cos^{-1}(0.4) + \pi n \quad \text{and} \quad \theta = -\frac{1}{2} \cos^{-1}(0.4) + \pi n \] 3. **Combine the Solutions:** The general solution can be concisely written as: \[ \theta = \pm \frac{1}{2} \cos^{-1}(0.4) + \pi n \quad \text{where} \quad n \in \mathbb{Z} \] **Alternatively, in degrees:** If you prefer to express the solution in degrees, calculate \( \cos^{-1}(0.4) \): \[ \cos^{-1}(0.4) \approx 66.42^\circ \] Thus, the general solution becomes: \[ \theta = \pm 33.21^\circ + 180^\circ n \quad \text{where} \quad n \in \mathbb{Z} \] --- **(b) \( 5 \sin \theta + 14 \cos \theta = 0 \)** **Solution:** To solve \( 5 \sin \theta + 14 \cos \theta = 0 \), follow these steps: 1. **Rearrange the Equation:** \[ 5 \sin \theta = -14 \cos \theta \] 2. **Divide Both Sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):** \[ 5 \tan \theta = -14 \] \[ \tan \theta = -\frac{14}{5} \] 3. **Find the General Solution:** The general solution for \( \tan \theta = -\frac{14}{5} \) is: \[ \theta = \tan^{-1}\left(-\frac{14}{5}\right) + \pi n \quad \text{where} \quad n \in \mathbb{Z} \] **Note:** \( \tan^{-1}\left(-\frac{14}{5}\right) \) can be calculated using a calculator to find the specific angles if needed. --- ### **Problem 8** **Solve for \( x \) if \( \sin \left(x + 30^\circ\right) = -0.2 \) where \( x \in \left(-180^\circ; 0^\circ\right) \)** **Solution:** To solve the equation \( \sin(x + 30^\circ) = -0.2 \) within the interval \( x \in (-180^\circ, 0^\circ) \), follow these steps: 1. **Let \( y = x + 30^\circ \).** Then, the equation becomes: \[ \sin y = -0.2 \] 2. **Find the General Solutions for \( y \):** The general solutions for \( \sin y = -0.2 \) are: \[ y = \sin^{-1}(-0.2) + 360^\circ n \quad \text{and} \quad y = 180^\circ - \sin^{-1}(-0.2) + 360^\circ n \quad \text{where} \quad n \in \mathbb{Z} \] Calculate \( \sin^{-1}(-0.2) \): \[ \sin^{-1}(-0.2) \approx -11.54^\circ \] Therefore, the solutions are: \[ y = -11.54^\circ + 360^\circ n \] \[ y = 180^\circ + 11.54^\circ + 360^\circ n = 191.54^\circ + 360^\circ n \] 3. **Express \( x \) in Terms of \( y \):** Since \( y = x + 30^\circ \), then: \[ x = y - 30^\circ \] Substituting the solutions for \( y \): - From \( y = -11.54^\circ + 360^\circ n \): \[ x = -11.54^\circ - 30^\circ + 360^\circ n = -41.54^\circ + 360^\circ n \] - From \( y = 191.54^\circ + 360^\circ n \): \[ x = 191.54^\circ - 30^\circ + 360^\circ n = 161.54^\circ + 360^\circ n \] 4. **Determine the Values of \( x \) Within \( (-180^\circ, 0^\circ) \):** - **First Solution:** \( x = -41.54^\circ + 360^\circ n \) - For \( n = 0 \): \[ x = -41.54^\circ \quad (\text{which is within } (-180^\circ, 0^\circ)) \] - For \( n = 1 \) or \( n = -1 \): \[ x = 318.46^\circ \quad \text{or} \quad x = -401.54^\circ \quad (\text{both are outside the interval}) \] - **Second Solution:** \( x = 161.54^\circ + 360^\circ n \) - For \( n = -1 \): \[ x = 161.54^\circ - 360^\circ = -198.46^\circ \quad (\text{which is outside } (-180^\circ, 0^\circ)) \] - For \( n = 0 \) or \( n = 1 \): \[ x = 161.54^\circ \quad \text{or} \quad x = 521.54^\circ \quad (\text{both are outside the interval}) \] 5. **Final Solution:** The only solution within \( (-180^\circ, 0^\circ) \) is: \[ x \approx -41.54^\circ \] **Rounded to one decimal place:** \[ x \approx -41.5^\circ \] --- ### **Summary of Solutions** - **Problem 7(a):** \[ \theta = \pm \frac{1}{2} \cos^{-1}(0.4) + \pi n \quad \text{for any integer } n \] *Alternatively in degrees:* \[ \theta = \pm 33.21^\circ + 180^\circ n \quad \text{for any integer } n \] - **Problem 7(b):** \[ \theta = \tan^{-1}\left(-\frac{14}{5}\right) + \pi n \quad \text{for any integer } n \] - **Problem 8:** \[ x \approx -41.5^\circ \]