If `0^@ lt theta lt 90^@`, then all the trigonometric ratios can be obtained when

Options

A. Only `sin theta` is given
B. Only `cos theta` is given
C. Only `tan theta`is given
D. any one of the six ratios is given

Answer

To solve the question, we need to determine under what conditions we can derive all six trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) when the angle θ lies between 0° and 90°.

### Step-by-Step Solution:

1. **Understanding Trigonometric Ratios**:
The six primary trigonometric ratios are:
- \( \sin \theta \)
- \( \cos \theta \)
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
- \( \csc \theta = \frac{1}{\sin \theta} \)
- \( \sec \theta = \frac{1}{\cos \theta} \)
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

2. **Analyzing Each Option**:
- **Option 1**: If only \( \sin \theta \) is given, we can find \( \csc \theta \) directly, but we cannot find \( \cos \theta \) or \( \tan \theta \) without additional information. Thus, this option is insufficient.
- **Option 2**: If only \( \cos \theta \) is given, we can find \( \sec \theta \) directly, but we cannot find \( \sin \theta \) or \( \tan \theta \) without additional information. Thus, this option is also insufficient.
- **Option 3**: If only \( \tan \theta \) is given, we can find \( \sin \theta \) and \( \cos \theta \) using the definition of tangent, but we cannot find \( \csc \theta \) or \( \sec \theta \) without additional information. Thus, this option is insufficient as well.
- **Option 4**: If any one of the six ratios is given, we can derive the others using the relationships between them. For example, if \( \sin \theta \) is known, we can find \( \cos \theta \) using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Similarly, if \( \tan \theta \) is known, we can derive \( \sin \theta \) and \( \cos \theta \) from it.

3. **Conclusion**:
The correct answer is **Option 4**: Any one of the six ratios is given, we can find all other ratios.