Question

If \( \tan \theta = 2 \) and \( \theta \) lies in the third quadrant, then the value of \( \sec \theta \) is

• A. \( -\sqrt{5} \)
• B. \( \sqrt{3} \)
• C. \( -\sqrt{2} \)
• D. \( \sqrt{5} \)

Hint:

In trigonometry, use the quadrant information to determine the correct sign of trigonometric functions. In the third quadrant, both \( \sin \theta \) and \( \cos \theta \) are negative.

Answer 1

The Correct Option is A

Step 1: Use the identity for \( \sec \theta \).
We know that: \[ \sec^2 \theta = 1 + \tan^2 \theta \] Substitute \( \tan \theta = 2 \) into the equation: \[ \sec^2 \theta = 1 + 2^2 = 1 + 4 = 5 \] Thus, \[ \sec \theta = \pm \sqrt{5} \]
Step 2: Determine the correct sign.
Since \( \theta \) lies in the third quadrant, both \( \sin \theta \) and \( \cos \theta \) are negative, so \( \sec \theta \) must also be negative. Therefore, \[ \sec \theta = -\sqrt{5} \]
Step 3: Conclusion.
Thus, the value of \( \sec \theta \) is \( -\sqrt{5} \), corresponding to option (A).