Question

The general solution of $ x $ satisfying the equation $ \sqrt{3} \sin x + \cos x = \sqrt{3} $, is given by

• A. \( x = n\pi \pm \frac{\pi}{3} \)
• B. \( x = n\pi \pm \frac{\pi}{6} \)
• C. \( x = n\pi \pm (-1)^n \frac{\pi}{3} \)
• D. \( x = n\pi \pm (-1)^n \frac{\pi}{4} \)

Hint:

When solving trigonometric equations, standard values like \( \frac{\pi}{3} \) are useful for simplifying the equation and determining the general solution.

Answer 1

The Correct Option is A

We are given the equation: \[ \sqrt{3} \sin x + \cos x = \sqrt{3} \]
Step 1: Rearrange the equation
Rearrange the given equation to make it more manageable: \[ \sqrt{3} \sin x = \sqrt{3} - \cos x \]
Step 2: Divide through by \( \sqrt{3} \)
This simplifies to: \[ \sin x = 1 - \frac{\cos x}{\sqrt{3}} \]
Step 3: Use standard trigonometric identities
We recognize that the sine function \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \), and by solving for \( x \), we find the general solutions: \[ x = n\pi \pm \frac{\pi}{3} \] This comes from the standard solution for trigonometric equations where the angles repeat every \( \pi \), and the general solution involves an addition or subtraction of \( \frac{\pi}{3} \).
Step 4: Conclusion
Thus, the correct general solution is \( x = n\pi \pm \frac{\pi}{3} \), which corresponds to option (a).