Geometry

Question Analysis  

The question asks which quadrant has both sin and cos negative, focusing on trigonometric sign conventions across quadrants.  

 

Key Concept Explanation  

Signs by quadrant (via ASTC rule):  

Q1: Sin (+), Cos (+)  

Q2: Sin (+), Cos (-)  

Q3: Sin (-), Cos (-)  

Q4: Sin (-), Cos (+)  

 

Step-by-Step Solution  

1. Recall the signs of sin (related to ) and cos (related to ):  

Sin < 0: y < 0 (Q3 or Q4).  

Cos < 0: x < 0 (Q2 or Q3).  

2. Find the overlap: Both x < 0 and y < 0 occur only in Quadrant III.  

 

Option Analysis  

Quadrant I: Both positive(Incorrect)

Quadrant II: Sin (+), Cos (-)(Incorrect)

Quadrant III: Both negative(Correct)

Quadrant IV: Sin (-), Cos (+)(Incorrect)  

 

Common Mistakes  

Misapplying the ASTC rule (e.g., confusing Q3 with Q4).  

Forgetting that tan is positive in Q3 (but the question only asks about sin/cos).  

 

Summary  

Both sin and cos are negative in Quadrant III.  

Problem-Solving Tips: 

Use the mnemonic ASTC (All Students Take Calculus) to remember:  

All (+) in Q1, Sin (+) in Q2, Tan (+) in Q3, Cos (+) in Q4.  

Relate to coordinates:

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