If `sin theta =40//41`, then `cot theta ` is
A. 40/9
B. 9/40
C. 9/41
D. 41/9

Options

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

Answer

To find \( \cot \theta \) given that \( \sin \theta = \frac{40}{41} \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Given \( \sin \theta = \frac{40}{41} \), we can calculate \( \sin^2 \theta \):
\[ \sin^2 \theta = \left(\frac{40}{41}\right)^2 = \frac{1600}{1681} \]

### Step 2: Calculate \( \cos^2 \theta \)
Now, we can find \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1600}{1681} = \frac{1681 - 1600}{1681} = \frac{81}{1681} \]

### Step 3: Calculate \( \cos \theta \)
Taking the square root to find \( \cos \theta \):
\[ \cos \theta = \sqrt{\frac{81}{1681}} = \frac{9}{41} \]

### Step 4: Calculate \( \cot \theta \)
Now, we can find \( \cot \theta \) using the definition:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{9}{41}}{\frac{40}{41}} = \frac{9}{40} \]

### Conclusion
Thus, the value of \( \cot \theta \) is \( \frac{9}{40} \), which corresponds to option B.