Question

If \(\sec \theta + \tan \theta = \frac{1}{3}\), then the quadrant in which \(2\theta\) lies is:

• A. 1st quadrant
• B. 2nd quadrant
• C. 3rd quadrant
• D. 4th quadrant

Hint:

When dealing with trigonometric identities and their implications on angle measurements, visualize the unit circle and the signs of the trigonometric functions in each quadrant to aid in determining the correct angle locations.

Answer 1

The Correct Option is C

Step 1: Analyze the condition \(\sec \theta + \tan \theta = \frac{1}{3}\).
- This identity suggests that both \(\sec \theta\) and \(\tan \theta\) are negative, indicating \(\theta\) is in the 4th quadrant. 
Step 2: Determine the quadrant of \(2\theta\).
- Given \(\theta\) in the 4th quadrant, \(2\theta\) will lie in the 3rd quadrant because adding \(360^\circ\) to \(\theta\) from the 4th quadrant places \(2\theta\) in the 3rd quadrant (as it ranges from \(270^\circ\) to \(360^\circ\)).