Which of the following is incorrect?

Options

A. `tan theta = 3 `
B. `sin theta = 3`
C. `sec theta = 3 `
D. `cot theta = 3 `

Answer

To determine which of the given options is incorrect regarding the values of trigonometric functions, let's analyze each option step by step.

### Step 1: Understand the ranges of trigonometric functions
- The range of \( \sin \theta \) is \([-1, 1]\).
- The range of \( \cos \theta \) is \([-1, 1]\).
- The range of \( \tan \theta \) is \((- \infty, + \infty)\).
- The range of \( \sec \theta \) is \((-\infty, -1] \cup [1, +\infty)\).
- The range of \( \csc \theta \) is \((-\infty, -1] \cup [1, +\infty)\).
- The range of \( \cot \theta \) is \((- \infty, + \infty)\).

### Step 2: Analyze each option
1. **Option 1: \( \tan \theta = 3 \)**
- Since \( \tan \theta \) can take any value in \((- \infty, + \infty)\), \( \tan \theta = 3 \) is possible.
- **Conclusion**: This option is correct.

2. **Option 2: \( \sin \theta = 3 \)**
- Since the range of \( \sin \theta \) is \([-1, 1]\), \( \sin \theta = 3 \) is not possible.
- **Conclusion**: This option is incorrect.

3. **Option 3: \( \sec \theta = 3 \)**
- Since \( \sec \theta = \frac{1}{\cos \theta} \), we can set \( \cos \theta = \frac{1}{3} \).
- The value \( \frac{1}{3} \) lies within the range of \( \cos \theta \) which is \([-1, 1]\).
- **Conclusion**: This option is correct.

4. **Option 4: \( \cot \theta = 3 \)**
- Since \( \cot \theta = \frac{1}{\tan \theta} \), we can set \( \tan \theta = \frac{1}{3} \).
- The value \( \frac{1}{3} \) is within the range of \( \tan \theta \) which is \((- \infty, + \infty)\).
- **Conclusion**: This option is correct.

### Final Conclusion
After analyzing all the options, we find that:
- **Option 2: \( \sin \theta = 3 \)** is the only incorrect statement.