Solve the equation \( 4 \cos x+2=3 \) for \( 0^{\circ} \leq x \leq 360^{\circ} \)

To solve the equation \( 4 \cos x + 2 = 3 \) for \( 0^{\circ} \leq x \leq 360^{\circ} \), follow these steps: ### Step 1: Isolate the Cosine Term Start by isolating the cosine term. \[ 4 \cos x + 2 = 3 \] Subtract 2 from both sides: \[ 4 \cos x = 1 \] Divide both sides by 4: \[ \cos x = \frac{1}{4} \] ### Step 2: Find the Principal Angle Find the principal angle whose cosine is \( \frac{1}{4} \). \[ x = \cos^{-1}\left(\frac{1}{4}\right) \] Using a calculator: \[ x \approx 75.52^{\circ} \] ### Step 3: Determine All Solutions in the Given Interval The cosine function is positive in the **first** and **fourth** quadrants. Therefore, there are two solutions within \( 0^{\circ} \leq x \leq 360^{\circ} \): 1. **First Quadrant:** \[ x \approx 75.52^{\circ} \] 2. **Fourth Quadrant:** \[ x = 360^{\circ} - 75.52^{\circ} = 284.48^{\circ} \] ### Final Answer The solutions to the equation \( 4 \cos x + 2 = 3 \) within the interval \( 0^{\circ} \leq x \leq 360^{\circ} \) are: \[ x \approx 75.52^{\circ} \quad \text{and} \quad x \approx 284.48^{\circ} \]