if `sinA=1/2`, then the value of `cot A`

Options

A. `sqrt(3)`
B. `1/sqrt(3)`
C. `sqrt(3)/2`
D. 1

Explanation

To find the value of \( \cot A \) given that \( \sin A = \frac{1}{2} \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sin^2 A + \cos^2 A = 1 \]
Given \( \sin A = \frac{1}{2} \), we can find \( \cos A \).

### Step 2: Calculate \( \cos A \)
Substituting \( \sin A \) into the Pythagorean identity:
\[ \left(\frac{1}{2}\right)^2 + \cos^2 A = 1 \]
This simplifies to:
\[ \frac{1}{4} + \cos^2 A = 1 \]
Subtract \( \frac{1}{4} \) from both sides:
\[ \cos^2 A = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 3: Take the Square Root
Now, take the square root to find \( \cos A \):
\[ \cos A = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]
(Note: We consider the positive root assuming \( A \) is in the first quadrant where both sine and cosine are positive.)

### Step 4: Calculate \( \cot A \)
Now, we can find \( \cot A \) using the definition:
\[ \cot A = \frac{\cos A}{\sin A} \]
Substituting the values we found:
\[ \cot A = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3} \]

### Final Answer
Thus, the value of \( \cot A \) is:
\[ \cot A = \sqrt{3} \]

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