Geometry

Question Analysis  

The question asks which quadrant has both cos and tan negative, requiring analysis of trigonometric sign conventions in each quadrant.  

 

Key Concept Explanation  

Signs by quadrant (ASTC rule):  

Q1: Cos (+), Tan (+)  

Q2: Cos (-), Tan (-)  

Q3: Cos (-), Tan (+)  

Q4: Cos (+), Tan (-)  

Tan = Sin/Cos, so its sign depends on the ratio of Sin and Cos.  

 

Step-by-Step Solution  

1. Analyze each quadrant:  

Q1: Both Cos and Tan are positive(Incorrect)

Q2: Cos (-), Tan = Sin (+)/Cos (-) = negative(Correct)

Q3: Cos (-), Tan = Sin (-)/Cos (-) = positive(Incorrect)

Q4: Cos (+), Tan = Sin (-)/Cos (+) = negative(Incorrect)

2. Only in Quadrant II are both Cos and Tan negative.  

 

Common Mistakes  

Confusing Tan’s sign in Q2 (e.g., forgetting Tan = Sin/Cos and Sin is positive in Q2).  

Misapplying the ASTC rule (e.g., assuming Q3 has negative Tan).  

 

Summary  

Both cos and tan are negative in Quadrant II.  

Problem-Solving Tips: 

Use the mnemonic ASTC (All Students Take Calculus) to recall positive functions:  

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

For Tan, use the ratio

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