Question

The general solution of the equation \( \cos(2x) = \frac{1}{2} \) is:

• A. \( x = \frac{\pi}{6} + n\pi \)
• B. \( x = \pm\frac{\pi}{3} + n\pi \)
• C. \( x = \frac{\pi}{6} + 2n\pi \)
• D. \( x = \pm\frac{\pi}{6} + n\pi \)

Hint:

Key Fact: Use identities and reduce to standard form: \( \cos \theta = \frac{1}{2} \Rightarrow \theta = \pm \frac{\pi}{3} + 2n\pi \)

Answer 1

The Correct Option is D

The equation \(\cos(2x) = \frac{1}{2}\) can be solved using the known values where the cosine function equals \(\frac{1}{2}\). 

According to trigonometric identities, \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) and \(\cos(-\frac{\pi}{3}) = \frac{1}{2}\). 

Thus, the angles where \(\cos(2x) = \frac{1}{2}\) are given by:

\[ 2x = \pm\frac{\pi}{3} + 2n\pi \]

solving for \(x\), we get:

\[ x = \pm\frac{\pi}{6} + n\pi \]

where \(n\) is any integer. Hence, the general solution to the equation is:

\( x = \pm\frac{\pi}{6} + n\pi \)

This matches the option \( x = \pm\frac{\pi}{6} + n\pi \).