1. Solve \( 7 \tan 2 \theta=6 \) for \( 0^{\circ} \leq \theta<360^{\circ} \)

To solve the equation \( 7 \tan 2\theta = 6 \), we start by isolating \( \tan 2\theta \): \[ \tan 2\theta = \frac{6}{7} \] Next, we need to find the angle \( 2\theta \) whose tangent is \( \frac{6}{7} \). We can use the inverse tangent function: \[ 2\theta = \tan^{-1}\left(\frac{6}{7}\right) \] Using a calculator, we find: \[ 2\theta \approx 40.60^\circ \quad \text{and} \quad 40.60^\circ + 180^\circ k \quad (k \in \mathbb{Z}) \] Now, to find angles within the range of \( 0^\circ \leq 2\theta < 720^\circ \): 1. For \( k = 0 \): \[ 2\theta \approx 40.60^\circ \] 2. For \( k = 1 \): \[ 2\theta \approx 40.60^\circ + 180^\circ = 220.60^\circ \] 3. For \( k = 2 \): \[ 2\theta \approx 40.60^\circ + 360^\circ = 400.60^\circ \quad \text{(out of range)} \] 4. For \( k = 1 \): \[ 2\theta \approx 220.60^\circ + 180^\circ = 400.60^\circ \quad \text{(out of range)} \] Thus, the angles for \( 2\theta \) within the specified range are \( 40.60^\circ \) and \( 220.60^\circ \). Now, we divide each by 2 to find \( \theta \): 1. For \( 2\theta \approx 40.60^\circ \): \[ \theta \approx \frac{40.60^\circ}{2} \approx 20.30^\circ \] 2. For \( 2\theta \approx 220.60^\circ \): \[ \theta \approx \frac{220.60^\circ}{2} \approx 110.30^\circ \] So, the solutions for \( \theta \) in the range \( 0^\circ \leq \theta < 360^\circ \) are: \[ \theta \approx 20.30^\circ, \quad 110.30^\circ \]