The value of `theta(0le theta le pi//2)` stisfying the equation `sin^(2)theta-2costheta+(1)/(4)=0` is :

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the equation \( \sin^2 \theta - 2 \cos \theta + \frac{1}{4} = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \sin^2 \theta = 1 - \cos^2 \theta \). Substituting this into the equation gives us:
\[ 1 - \cos^2 \theta - 2 \cos \theta + \frac{1}{4} = 0 \]

### Step 2: Rearrange the equation
Rearranging the equation, we have:
\[ -\cos^2 \theta - 2 \cos \theta + 1 + \frac{1}{4} = 0 \]
This simplifies to:
\[ -\cos^2 \theta - 2 \cos \theta + \frac{5}{4} = 0 \]

### Step 3: Multiply through by -4
To eliminate the fraction, we multiply the entire equation by -4:
\[ 4 \cos^2 \theta + 8 \cos \theta - 5 = 0 \]

### Step 4: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ (2 \cos \theta - 1)(2 \cos \theta + 5) = 0 \]

### Step 5: Solve for \( \cos \theta \)
Setting each factor to zero gives us two equations:
1. \( 2 \cos \theta - 1 = 0 \)
2. \( 2 \cos \theta + 5 = 0 \)

From the first equation:
\[ 2 \cos \theta = 1 \implies \cos \theta = \frac{1}{2} \]

From the second equation:
\[ 2 \cos \theta = -5 \implies \cos \theta = -\frac{5}{2} \]
Since the value of \( \cos \theta \) must be between -1 and 1, we discard \( \cos \theta = -\frac{5}{2} \).

### Step 6: Find \( \theta \)
Now, we know \( \cos \theta = \frac{1}{2} \). The angle \( \theta \) that satisfies this in the range \( 0 \leq \theta \leq \frac{\pi}{2} \) is:
\[ \theta = \frac{\pi}{3} \]

### Final Answer
Thus, the value of \( \theta \) satisfying the equation is:
\[ \theta = \frac{\pi}{3} \]