Which of trigonometric ratios are positive for the angles,
(i) `210^(@)` (ii) `-405^(@)`

Answer

To determine which trigonometric ratios are positive for the angles \( 210^\circ \) and \( -405^\circ \), we will analyze each angle based on the unit circle and the properties of trigonometric functions in different quadrants.

### Step-by-Step Solution:

**Step 1: Analyze the angle \( 210^\circ \)**

1. **Locate \( 210^\circ \) on the unit circle:**
- The angle \( 210^\circ \) is in the third quadrant (between \( 180^\circ \) and \( 270^\circ \)).

2. **Identify the positive trigonometric ratios in the third quadrant:**
- In the third quadrant, the only positive trigonometric ratios are:
- Tangent (\( \tan \))
- Cotangent (\( \cot \))

**Conclusion for \( 210^\circ \):**
- The positive trigonometric ratios are \( \tan \) and \( \cot \).

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**Step 2: Analyze the angle \( -405^\circ \)**

1. **Convert \( -405^\circ \) to a positive angle:**
- To find the equivalent positive angle, we can add \( 360^\circ \) to \( -405^\circ \):
\[ -405^\circ + 360^\circ = -45^\circ \]
- Adding \( 360^\circ \) again gives:
\[ -45^\circ + 360^\circ = 315^\circ \]
- Therefore, \( -405^\circ \) is equivalent to \( 315^\circ \).

2. **Locate \( 315^\circ \) on the unit circle:**
- The angle \( 315^\circ \) is in the fourth quadrant (between \( 270^\circ \) and \( 360^\circ \)).

3. **Identify the positive trigonometric ratios in the fourth quadrant:**
- In the fourth quadrant, the only positive trigonometric ratios are:
- Cosine (\( \cos \))
- Secant (\( \sec \))

**Conclusion for \( -405^\circ \):**
- The positive trigonometric ratios are \( \cos \) and \( \sec \).

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### Final Answers:
- For \( 210^\circ \): Positive ratios are \( \tan \) and \( \cot \).
- For \( -405^\circ \): Positive ratios are \( \cos \) and \( \sec \).

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