If sin A = `(1)/(2)` , then find the value of cos A.

Answer

To find the value of cos A when sin A = \( \frac{1}{2} \), we can use the Pythagorean identity in trigonometry, which states that:

\[ \sin^2 A + \cos^2 A = 1 \]

### Step 1: Substitute the value of sin A
Since we know that \( \sin A = \frac{1}{2} \), we can substitute this value into the Pythagorean identity.

\[ \left(\frac{1}{2}\right)^2 + \cos^2 A = 1 \]

### Step 2: Calculate \( \sin^2 A \)
Now, calculate \( \left(\frac{1}{2}\right)^2 \):

\[ \frac{1}{4} + \cos^2 A = 1 \]

### Step 3: Isolate \( \cos^2 A \)
Next, we need to isolate \( \cos^2 A \) by subtracting \( \frac{1}{4} \) from both sides:

\[ \cos^2 A = 1 - \frac{1}{4} \]

### Step 4: Simplify the right side
Now, simplify the right side:

\[ \cos^2 A = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]

### Step 5: Take the square root
To find \( \cos A \), we take the square root of both sides. Remember that cosine can be positive or negative depending on the angle A, but we will consider the principal value first:

\[ \cos A = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \( \cos A \) is:

\[ \cos A = \frac{\sqrt{3}}{2} \]