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

To solve the equation \( 7 \tan 2\theta = 6 \) for \( 0^{\circ} \leq \theta < 360^{\circ} \), follow these steps: 1. **Isolate the Tangent Function:** \[ 7 \tan 2\theta = 6 \implies \tan 2\theta = \frac{6}{7} \] 2. **Find the General Solution for \( 2\theta \):** \[ 2\theta = \tan^{-1}\left(\frac{6}{7}\right) + k \cdot 180^{\circ} \quad \text{for integer } k \] Here, \( \tan^{-1}\left(\frac{6}{7}\right) \) is the principal angle whose tangent is \( \frac{6}{7} \). 3. **Calculate \( \tan^{-1}\left(\frac{6}{7}\right) \) Numerically:** \[ \tan^{-1}\left(\frac{6}{7}\right) \approx 40.62^{\circ} \] 4. **Solve for \( \theta \):** \[ \theta = \frac{40.62^{\circ} + k \cdot 180^{\circ}}{2} = 20.31^{\circ} + k \cdot 90^{\circ} \] 5. **Determine All Solutions within the Given Range:** We need \( 0^{\circ} \leq \theta < 360^{\circ} \). Plugging in integer values for \( k \): - For \( k = 0 \): \( \theta = 20.31^{\circ} \) - For \( k = 1 \): \( \theta = 110.31^{\circ} \) - For \( k = 2 \): \( \theta = 200.31^{\circ} \) - For \( k = 3 \): \( \theta = 290.31^{\circ} \) \( k = 4 \) would yield \( \theta = 380.31^{\circ} \), which is outside the desired range. **Final Solutions:** \[ \theta \approx 20.31^{\circ}, \quad 110.31^{\circ}, \quad 200.31^{\circ}, \quad \text{and} \quad 290.31^{\circ} \] **Rounded to Two Decimal Places:** \[ \theta \approx 20.31^{\circ},\ 110.31^{\circ},\ 200.31^{\circ},\ \text{and}\ 290.31^{\circ} \]