Question

The general solution of $2\cos\theta - \sqrt{3} = 0$ is}

• A. $\theta = n\pi + (-1)^n \frac{\pi}{6}$
• B. $\theta = 2n\pi \pm \frac{\pi}{6}$
• C. $\theta = 2n\pi \pm \frac{\pi}{3}$
• D. $\theta = n\pi + (-1)^n \frac{\pi}{3}$

Hint:

Remember:
• $\cos\theta = \cos\alpha \Rightarrow \theta = 2n\pi \pm \alpha$
• $\sin\theta = \sin\alpha \Rightarrow \theta = n\pi + (-1)^n\alpha$

Answer 1

The Correct Option is B

Step 1: Simplifying the given equation
We are given: \[ 2\cos\theta - \sqrt{3} = 0 \] First, isolate the trigonometric term: \[ 2\cos\theta = \sqrt{3} \] Now divide both sides by 2: \[ \cos\theta = \frac{\sqrt{3}}{2} \]

Step 2: Finding principal value
We know from standard trigonometric values that: \[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \] Thus, the principal solution angle is: \[ \theta = \frac{\pi}{6} \]

Step 3: Applying general solution of cosine
For cosine equations, the general solution formula is: \[ \theta = 2n\pi \pm \alpha \] where $\alpha$ is the principal angle. Substituting $\alpha = \frac{\pi}{6}$: \[ \theta = 2n\pi \pm \frac{\pi}{6} \]

Step 4: Final interpretation
This represents all angles where cosine equals $\frac{\sqrt{3}}{2}$, including both symmetric positions in the unit circle. Thus, the correct answer is option (B).