2.4 Solve for \( \alpha \) if : \( 2 \tan \alpha=4 \) and \( \alpha \in\left[90^{\circ} ; 360^{\circ}\right] \)

To solve for \( \alpha \) in the equation \( 2 \tan \alpha = 4 \), first divide both sides by 2 to get \( \tan \alpha = 2 \). Next, you want to find the angles where the tangent is equal to 2. The basic angle is: \[ \alpha_0 = \tan^{-1}(2) \] Calculating this gives approximately \( \alpha_0 \approx 63.43^{\circ} \). However, since we are looking for solutions in the range \( [90^{\circ}, 360^{\circ}] \), we can find additional angles. The tangent function is positive in the third and first quadrants, but in our specified range, we focus on the third quadrant. The general solution for tangent is: \[ \alpha = \alpha_0 + n \cdot 180^{\circ} \] For the third quadrant, since \( n = 1 \): \[ \alpha = 63.43^{\circ} + 180^{\circ} \approx 243.43^{\circ} \] Thus, the solution for \( \alpha \) in the interval \( [90^{\circ}, 360^{\circ}] \) is: \[ \alpha \approx 243.43^{\circ} \] So, final answer: \( \alpha \approx 243.43^{\circ} \).