If `sin A = 3/4` calculate sec A.

Answer

To solve the problem of finding \( \sec A \) given that \( \sin A = \frac{3}{4} \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 A + \cos^2 A = 1 \]

### Step 2: Substitute the Value of \( \sin A \)
Given \( \sin A = \frac{3}{4} \), we can substitute this value into the identity:
\[ \left(\frac{3}{4}\right)^2 + \cos^2 A = 1 \]

### Step 3: Calculate \( \sin^2 A \)
Calculating \( \left(\frac{3}{4}\right)^2 \):
\[ \frac{9}{16} + \cos^2 A = 1 \]

### Step 4: Isolate \( \cos^2 A \)
Now, we need to isolate \( \cos^2 A \):
\[ \cos^2 A = 1 - \frac{9}{16} \]

### Step 5: Convert 1 to a Fraction
Convert 1 to a fraction with a denominator of 16:
\[ \cos^2 A = \frac{16}{16} - \frac{9}{16} = \frac{7}{16} \]

### Step 6: Take the Square Root to Find \( \cos A \)
Now, take the square root of both sides to find \( \cos A \):
\[ \cos A = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]

### Step 7: Calculate \( \sec A \)
The secant function is the reciprocal of the cosine function:
\[ \sec A = \frac{1}{\cos A} = \frac{1}{\frac{\sqrt{7}}{4}} = \frac{4}{\sqrt{7}} \]

### Final Answer
Thus, the value of \( \sec A \) is:
\[ \sec A = \frac{4}{\sqrt{7}} \]

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