Solve for x, where `0 le x le 90^(@) sin^(2) x + cos^(2) 30^(@) = (5)/(4)`

Answer

To solve the equation \( \sin^2 x + \cos^2 30^\circ = \frac{5}{4} \) for \( 0 \leq x \leq 90^\circ \), we can follow these steps:

### Step 1: Substitute the value of \( \cos 30^\circ \)
We know that \( \cos 30^\circ = \frac{\sqrt{3}}{2} \). Therefore, we can calculate \( \cos^2 30^\circ \):

\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 2: Rewrite the equation
Now substitute \( \cos^2 30^\circ \) back into the equation:

\[ \sin^2 x + \frac{3}{4} = \frac{5}{4} \]

### Step 3: Isolate \( \sin^2 x \)
Next, we can isolate \( \sin^2 x \) by subtracting \( \frac{3}{4} \) from both sides:

\[ \sin^2 x = \frac{5}{4} - \frac{3}{4} \]

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

\[ \sin^2 x = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Take the square root
Now, take the square root of both sides to solve for \( \sin x \):

\[ \sin x = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 6: Find the angle \( x \)
We know that \( \sin 45^\circ = \frac{1}{\sqrt{2}} \). Therefore:

\[ x = 45^\circ \]

### Final Answer
Thus, the solution for \( x \) is:

\[ x = 45^\circ \]