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

Answer

To find the value of sec A given that sin A = 3/4, we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:
\[ \cos^2 A + \sin^2 A = 1 \]
Given that \(\sin A = \frac{3}{4}\), we can substitute this value into the identity.

### Step 2: Calculate \(\sin^2 A\)
First, we calculate \(\sin^2 A\):
\[ \sin^2 A = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Step 3: Substitute into the Pythagorean Identity
Now, substitute \(\sin^2 A\) into the identity:
\[ \cos^2 A + \frac{9}{16} = 1 \]

### Step 4: Solve for \(\cos^2 A\)
Rearranging the equation gives:
\[ \cos^2 A = 1 - \frac{9}{16} \]
To perform the subtraction, convert 1 into a fraction with a denominator of 16:
\[ 1 = \frac{16}{16} \]
Now, subtract:
\[ \cos^2 A = \frac{16}{16} - \frac{9}{16} = \frac{7}{16} \]

### Step 5: Calculate \(\cos A\)
Taking the square root of both sides gives:
\[ \cos A = \pm \sqrt{\frac{7}{16}} = \pm \frac{\sqrt{7}}{4} \]

### Step 6: Find \(\sec A\)
The secant function is the reciprocal of the cosine function:
\[ \sec A = \frac{1}{\cos A} \]
Thus:
\[ \sec A = \frac{1}{\pm \frac{\sqrt{7}}{4}} = \pm \frac{4}{\sqrt{7}} \]

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

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