Tanθ is positive and Sinθ is negative. In which quadrant does θ lies

Analyze the question: We need the quadrant where tan θ > 0 and sin θ < 0 at the same time.

Concept: the ASTC (CAST) rule

• Quadrant 1: All ratios are positive
• Quadrant 2: Only Sine (and cosec) is positive
• Quadrant 3: Only Tangent (and cot) is positive
• Quadrant 4: Only Cosine (and sec) is positive

Solution:

1. Sin θ is negative, so θ is in Quadrant 3 or Quadrant 4 (sine is positive in Q1 and Q2).
2. Tan θ is positive, so θ is in Quadrant 1 or Quadrant 3 (tangent is positive where sine and cosine have the same sign).
3. The only quadrant satisfying both conditions is Quadrant 3.

Check: In the third quadrant, sin θ < 0 and cos θ < 0, so tan θ = sin θ / cos θ = (negative)/(negative) = positive. ✔

Answer: B. Third only

Exam tip: Remember "All Students Take Chocolate" (A-S-T-C), going anticlockwise from Quadrant 1. Whichever ratio is named in a quadrant is positive there, along with its reciprocal. For "two conditions" questions, list the quadrants for each condition and pick the one they share.