Which one of the following is positive in the third quadrant?

Options

A. `sin theta`
B. `cos theta`
C. `tan theta`
D. `sec theta`

Answer

To determine which trigonometric function is positive in the third quadrant, we can analyze the signs of the trigonometric functions in each quadrant.

### Step-by-step Solution:

1. **Identify the Quadrants**:
- The unit circle is divided into four quadrants:
- 1st Quadrant: 0° to 90° (All trigonometric functions are positive)
- 2nd Quadrant: 90° to 180° (Sine is positive, Cosine and Tangent are negative)
- 3rd Quadrant: 180° to 270° (Sine and Cosine are negative, Tangent is positive)
- 4th Quadrant: 270° to 360° (Cosine is positive, Sine and Tangent are negative)

2. **Analyze the Signs of Trigonometric Functions**:
- In the third quadrant (180° to 270°):
- Sine (sin θ) is negative.
- Cosine (cos θ) is negative.
- Tangent (tan θ) is positive (since tan θ = sin θ / cos θ).
- Cosecant (csc θ) is negative (since csc θ = 1/sin θ).
- Secant (sec θ) is negative (since sec θ = 1/cos θ).
- Cotangent (cot θ) is negative (since cot θ = 1/tan θ).

3. **Conclusion**:
- The only trigonometric function that is positive in the third quadrant is **Tangent (tan θ)**.

### Final Answer:
**Tangent (tan θ)** is positive in the third quadrant.

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