If `cot 52^(@) = b, tan 38^(@) = ? `
A.`sqrt(b)`
B. `sqrt(b)//2`
C . `- b`
D. b

Options

A. C
B. D
C. B
D. A

Answer

To solve the problem, we need to find the value of \( \tan 38^\circ \) given that \( \cot 52^\circ = b \).

### Step-by-Step Solution:

1. **Understanding the Relationship Between Cotangent and Tangent**:
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Therefore, if \( \cot 52^\circ = b \), we can express this in terms of tangent:
\[ b = \cot 52^\circ = \frac{1}{\tan 52^\circ} \]

2. **Using the Complementary Angle Identity**:
We also know the trigonometric identity:
\[ \tan(90^\circ - \theta) = \cot \theta \]
Applying this to our case:
\[ \tan(90^\circ - 52^\circ) = \tan 38^\circ \]
Thus:
\[ \tan 38^\circ = \cot 52^\circ \]

3. **Substituting the Value of b**:
Since we established that \( \tan 38^\circ = \cot 52^\circ \) and \( \cot 52^\circ = b \), we can substitute:
\[ \tan 38^\circ = b \]

4. **Conclusion**:
Therefore, the value of \( \tan 38^\circ \) is equal to \( b \).

### Final Answer:
The answer is \( D. b \).