If tan y is positive and sin y is negative, in which quadrant would y lie?

Step 1: Analyze the question We need the quadrant where tan y > 0 and sin y < 0 at the same time.

Step 2: Recall the concept (ASTC / "All Students Take Care" rule)

• 1st quadrant (0° to 90°): All ratios are positive
• 2nd quadrant (90° to 180°): Only Sine (and cosec) is positive
• 3rd quadrant (180° to 270°): Only Tangent (and cot) is positive
• 4th quadrant (270° to 360°): Only Cosine (and sec) is positive

Step 3: Apply the conditions

• sin y is negative, so y is in the 3rd or 4th quadrant (sine is positive in the 1st and 2nd).
• tan y is positive, so y is in the 1st or 3rd quadrant.
• The quadrant that satisfies both conditions is the 3rd quadrant.

Step 4: Check with an example Take y = 225°. sin 225° = −√2/2 (negative), cos 225° = −√2/2 (negative), so tan 225° = (−)/(−) = +1 (positive). This confirms it.

Answer: C. Third only

Tip: When two conditions are given, find the quadrants for each separately and pick the one they share. Remember "All Students Take Care" going anticlockwise from the 1st quadrant: All, Sin, Tan, Cos.