Question

The common principal solution of the equations $\sin \theta = -1/2$ and $\tan \theta = 1/\sqrt{3}$ is \dots}

• A. $\pi/6$
• B. $5\pi/6$
• C. $7\pi/6$
• D. $11\pi/6$

Hint:

The ASTC (All Students Take Calculus) mnemonic is foolproof for these problems. Quadrant 1 (All +), Quadrant 2 (Sin +), Quadrant 3 (Tan +), Quadrant 4 (Cos +). By combining the sign constraints, the quadrant isolates itself instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Identify the quadrant where $\sin \theta$ is negative and $\tan \theta$ is positive.

Step 2: Formula Application:
$\sin \theta < 0$ in III and IV quadrants. $\tan \theta > 0$ in I and III quadrants. The common quadrant is the III quadrant.

Step 3: Explanation:
The reference angle for $1/2$ and $1/\sqrt{3}$ is $\pi/6$. In the III quadrant, the angle is $\pi + \alpha$. $\theta = \pi + \pi/6 = 7\pi/6$.

Step 4: Final Answer:
The common principal solution is $7\pi/6$.