Question

If $\tan \theta + \sec \theta = \sqrt{3}$ then the principal value of $\theta$ in $[0, 2\pi]$ is

• A. $\pi/4$
• B. $\pi/6$
• C. $\pi/2$
• D. $2\pi/3$

Hint:

If $\sec\theta + \tan\theta = k$, then $\sec\theta - \tan\theta = 1/k$. This makes solving for $\theta$ much faster.

Answer 1

The Correct Option is B

Step 1: Concept
Use the identity $\sec^2 \theta - \tan^2 \theta = 1$ along with the given linear equation.

Step 2: Meaning
Since $(\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1$ and $\sec \theta + \tan \theta = \sqrt{3}$, it follows that $\sec \theta - \tan \theta = 1/\sqrt{3}$.

Step 3: Analysis
Subtracting the equations: $(\sec \theta + \tan \theta) - (\sec \theta - \tan \theta) = \sqrt{3} - 1/\sqrt{3} \implies 2\tan \theta = 2/\sqrt{3}$. Thus, $\tan \theta = 1/\sqrt{3}$.

Step 4: Conclusion
The principal value for $\tan \theta = 1/\sqrt{3}$ in the first quadrant is $\pi/6$.
Final Answer: (B)