Solve \( 4 \sin x-1=2 \) for \( 0^{\circ} \leq x \leq 360^{\circ} \)

To solve the equation \( 4 \sin x - 1 = 2 \) for \( 0^{\circ} \leq x \leq 360^{\circ} \), follow these steps: 1. **Isolate the sine term:** \[ 4 \sin x - 1 = 2 \\ 4 \sin x = 3 \\ \sin x = \frac{3}{4} \] 2. **Find the reference angle:** The reference angle \( x_{\text{ref}} \) is the angle whose sine is \( \frac{3}{4} \): \[ x_{\text{ref}} = \arcsin\left(\frac{3}{4}\right) \approx 48.59^{\circ} \] 3. **Determine all solutions within the given interval:** The sine function is positive in the **first** and **second** quadrants. - **First Quadrant Solution:** \[ x_1 \approx 48.59^{\circ} \] - **Second Quadrant Solution:** \[ x_2 = 180^{\circ} - x_{\text{ref}} \approx 180^{\circ} - 48.59^{\circ} = 131.41^{\circ} \] 4. **Final Solutions:** \[ x \approx 48.59^{\circ} \quad \text{and} \quad x \approx 131.41^{\circ} \] **Therefore, the solutions to the equation \( 4 \sin x - 1 = 2 \) within the interval \( 0^{\circ} \leq x \leq 360^{\circ} \) are:** \[ x \approx 48.59^{\circ} \quad \text{and} \quad x \approx 131.41^{\circ} \]