Question

The principal solutions of $\cos 2x = -\frac{1}{2}$ are

• A. $x = -\frac{2\pi}{3},\ \frac{4\pi}{3}$
• B. $x = \frac{\pi}{3},\ \frac{2\pi}{3}$
• C. $x = -\frac{\pi}{3},\ \frac{5\pi}{6}$
• D. $x = \frac{\pi}{3},\ \frac{7\pi}{6}$

Hint:

Always divide by the coefficient of $x$ after finding the general solutions of a trigonometric equation.

Answer 1

The Correct Option is B

Step 1: Understanding the cosine equation.
\[ \cos 2x = -\frac{1}{2} \] Cosine is negative in the second and third quadrants.
Step 2: Finding the reference angle.
\[ \cos \theta = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{3} \]
Step 3: Writing principal values for $2x$.
\[ 2x = \frac{2\pi}{3},\ \frac{4\pi}{3} \]
Step 4: Solving for $x$.
\[ x = \frac{\pi}{3},\ \frac{2\pi}{3} \]
Step 5: Conclusion.
The principal solutions are $x = \frac{\pi}{3}$ and $x = \frac{2\pi}{3}$.