If `3 sec^(2) x - 4 = 0,` then the value of `x ( 0 lt x lt 90^(@))`

Options

A. `45 ^(@)`
B. `15 ^(@)`
C. `30^(@)`
D. `60^(@)`

Answer

To solve the equation \(3 \sec^2 x - 4 = 0\) for \(x\) in the range \(0 < x < 90^\circ\), follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ 3 \sec^2 x - 4 = 0 \]
Add 4 to both sides:
\[ 3 \sec^2 x = 4 \]

### Step 2: Isolate \(\sec^2 x\)
Now, divide both sides by 3:
\[ \sec^2 x = \frac{4}{3} \]

### Step 3: Take the square root
To find \(\sec x\), take the square root of both sides:
\[ \sec x = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} \]

### Step 4: Convert secant to cosine
Recall that \(\sec x = \frac{1}{\cos x}\). Therefore, we can write:
\[ \frac{1}{\cos x} = \frac{2}{\sqrt{3}} \]
Taking the reciprocal gives:
\[ \cos x = \frac{\sqrt{3}}{2} \]

### Step 5: Find the angle \(x\)
Now, we need to determine the angle \(x\) such that \(\cos x = \frac{\sqrt{3}}{2}\). The angle that satisfies this in the range \(0 < x < 90^\circ\) is:
\[ x = 30^\circ \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{30^\circ} \]