If `3 cos^(2) x - 2 sin^(2) x = - 0.75` and `0^(@) le x le 90^(@)` , then x =

Options

A. `60^(@)`
B. `45^(@)`
C. `30^(@)`
D. `90^(@)`

Answer

To solve the equation \(3 \cos^2 x - 2 \sin^2 x = -0.75\) for \(0^\circ \leq x \leq 90^\circ\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3 \cos^2 x - 2 \sin^2 x = -0.75 \]
To make calculations easier, we can rearrange it:
\[ 3 \cos^2 x - 2 \sin^2 x + 0.75 = 0 \]

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \cos^2 x + \sin^2 x = 1 \]
We can express \(\cos^2 x\) in terms of \(\sin^2 x\):
\[ \cos^2 x = 1 - \sin^2 x \]

### Step 3: Substitute \(\cos^2 x\) in the equation
Substituting \(\cos^2 x\) into the equation gives:
\[ 3(1 - \sin^2 x) - 2 \sin^2 x + 0.75 = 0 \]
Expanding this:
\[ 3 - 3 \sin^2 x - 2 \sin^2 x + 0.75 = 0 \]
Combining like terms:
\[ 3 - 5 \sin^2 x + 0.75 = 0 \]
This simplifies to:
\[ 3.75 - 5 \sin^2 x = 0 \]

### Step 4: Isolate \(\sin^2 x\)
Rearranging gives:
\[ 5 \sin^2 x = 3.75 \]
Dividing both sides by 5:
\[ \sin^2 x = \frac{3.75}{5} = 0.75 \]

### Step 5: Solve for \(\sin x\)
Taking the square root of both sides:
\[ \sin x = \sqrt{0.75} = \frac{\sqrt{3}}{2} \]

### Step 6: Find the angle \(x\)
Now, we need to find \(x\) such that:
\[ \sin x = \frac{\sqrt{3}}{2} \]
In the range \(0^\circ \leq x \leq 90^\circ\), this occurs at:
\[ x = 60^\circ \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{60^\circ} \]